Finite Group Databasebeta

← #10820040

Subgroups — #10820040

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82348C211yesmaximal, characteristic
2C2×#5011C211yesmaximal, characteristic
3C2×C8×Dic6C211yesmaximal, characteristic, complemented
4C3×#1954C211yesmaximal, characteristic
5#70473C211yesmaximal, characteristic, complemented
6#70442C211yesmaximal, characteristic, complemented
7#69951C211yesmaximal, characteristic, complemented
8#69950C211yesmaximal, characteristic, complemented
9#69949C211yesmaximal, characteristic, complemented
10#69947C211yesmaximal, characteristic, complemented
11#69914C211yesmaximal, characteristic, complemented
12#69913C211yesmaximal, characteristic, complemented
13#69913C211yesmaximal, characteristic, complemented
14#69912C211yesmaximal, characteristic, complemented
15#69755C211yesmaximal, characteristic, complemented
16#23210—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C22×#208C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27#5222C411yescharacteristic
28#5222C411yescharacteristic
29C4×#217C2211yescharacteristic
30C4×#217C2211yescharacteristic
31C4×#217C2211yescharacteristic
32C4×#217C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#5011—24no—
44#4985C411yescharacteristic
45#4985C411yescharacteristic
46#4953C411yescharacteristic
47#4953C411yescharacteristic
48#4947C411yescharacteristic
49#4947C411yescharacteristic
50#4932C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52C2×#803C2211yescharacteristic
53C8×Dic6—24no—
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×C8×Q8—31no—
63C2×#435—31no—
64#2398—31no—
65#1985—31no—
66#1954D311yescharacteristic
67#1924—31no—
68#1923—31no—
69#1923—31no—
70#1923—31no—
71#1851—31no—
72#1850—31no—
73#1768—31no—
74#1768—31no—
75#1753—31no—
76C8×C22⋊C4—31no—
77C22×C3×D4C2×C411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2×C411yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C2×C3×C4⋊C4C2×C411yescharacteristic
88C2×C3×C22⋊C4C2×C411yescharacteristic
89C2×C3×C22⋊C4C2×C411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217C2×C411yescharacteristic
92C2×#217C2×C411yescharacteristic
93C2×#217C2×C411yescharacteristic
94C2×#217C2×C411yescharacteristic
95C2×#217C2×C411yescharacteristic
96C2×#217C2×C411yescharacteristic
97C2×#217C2×C411yescharacteristic
98C2×#217C2×C411yescharacteristic
99C2×C4⋊Dic3C2×C411yescharacteristic
100C2×Dic3⋊C4C2×C411yescharacteristic
101C2×Dic3⋊C4C2×C411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×#208—22no—
107C2×#208—22no—
108C2×#208—41no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—41no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C4×Dic6—22no—
117C4×Dic6—22no—
118C3×C4⋊C8—22no—
119C3×C22⋊C8—22no—
120C3×C22⋊C8—22no—
121C3×C22⋊C8—41no—
122C3×C22⋊C8—41no—
123C3×C4×C8—22no—
124C3⋊#95C2×C411yescharacteristic
125C3⋊#95C2×C411yescharacteristic
126C3⋊#95C2×C411yescharacteristic
127C3⋊#95C2×C411yescharacteristic
128C3⋊#95C2×C411yescharacteristic
129C3⋊#95C2×C411yescharacteristic
130C3⋊#95C2×C411yescharacteristic
131C3⋊#95C2×C411yescharacteristic
132C3⋊#95C2×C411yescharacteristic
133C3⋊#95C2×C411yescharacteristic
134C3⋊#95C2×C411yescharacteristic
135C3⋊#95C2×C411yescharacteristic
136#830—22no—
137#830—22no—
138#815—41no—
139Dic3⋊C8—22no—
140Dic3⋊C8—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144C8×Dic3—22no—
145C8×Dic3—41no—
146C4⋊(C3⋊C8)D414yes—
147#803—22no—
148C22×M16—31no—
149C23×C8—31no—
150C2×C4×Q8—31no—
151C2×C4×D4C2×D311yescharacteristic
152C8×Q8—64no—
153#435—64no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×C4⋊C8—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×C22⋊C8—31no—
161C2×C22⋊C8—31no—
162C2×C22⋊C8C2×D311yescharacteristic
163C2×C22⋊C8C2×D311yescharacteristic
164C2×C22⋊C8C2×D311yescharacteristic
165C2×C22⋊C8C2×D311yescharacteristic
166C2×#97—31no—
167C2×#97—31no—
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171C2×C4×C8C2×D311yescharacteristic
172#400—31no—
173#400—31no—
174#397—31no—
175#397—31no—
176#394—31no—
177#394—31no—
178#393—31no—
179#393—31no—
180C4×C22⋊C4—31no—
181C4×C22⋊C4—31no—
182C4×C22⋊C4—31no—
183C4×C22⋊C4—31no—
184#376—31no—
185#336—31no—
186#336—31no—
187#336—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193C24×C3C22×C411yescharacteristic
194C24×C3C22×C411yescharacteristic
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—41no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C22×C411yescharacteristic
211C22×C3×C4C22×C411yescharacteristic
212C22×C3×C4C22×C411yescharacteristic
213C22×C3×C4C22×C411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3C22×C411yescharacteristic
216C22×Dic3C22×C411yescharacteristic
217C22×Dic3C22×C411yescharacteristic
218C22×Dic3C22×C411yescharacteristic
219C22×Dic3C22×C411yescharacteristic
220C22×Dic3C22×C411yescharacteristic
221C22×Dic3C22×C411yescharacteristic
222C22×Dic3C22×C411yescharacteristic
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8#4112yes—
238C3×C4⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C42C2×D412yes—
242C3×C42#4112yes—
243#217—22no—
244#217—22no—
245#217—22no—
246#217—22no—
247#217—22no—
248#217—22no—
249#217—41no—
250#217—41no—
251C4⋊Dic3—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262#208—44no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—44no—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8C2×D412yes—
275C2×C3⋊C8#4112yes—
276C22×Q8—31no—
277C22×D4C4×D311yescharacteristic
278C23×C4C22×D311yescharacteristic
279C23×C4C22×D311yescharacteristic
280C2×M16—62no—
281C2×M16—62no—
282C2×M16—121no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—62no—
288C22×C8—62no—
289C22×C8—121no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4×Q8—62no—
295C4×Q8—62no—
296C4×D4—22no—
297C4×D4—22no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4C4×D311yescharacteristic
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4C4×D311yescharacteristic
311C2×C22⋊C4C4×D311yescharacteristic
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42C22×D311yescharacteristic
317C4⋊C8—22no—
318C4⋊C8—34no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321C4⋊C8—62no—
322C4⋊C8—62no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—41no—
326C22⋊C8—41no—
327C22⋊C8—62no—
328C22⋊C8—62no—
329#97—62no—
330#97—121no—
331C4×C8—22no—
332C4×C8—62no—
333C4×C8—62no—
334C4×C8—121no—
335#95—31no—
336#95—31no—
337#95—31no—
338#95—31no—
339#95—31no—
340#95—31no—
341#95—31no—
342#95—31no—
343#95—31no—
344#95—31no—
345#95—31no—
346#95—31no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3#13112yes—
361C23×C3C23×C411yescharacteristic
362C3×D4—44no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4C4×D412yes—
382C2×C3×C4C4×D412yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3#13112yes—
412C2×Dic3#13112yes—
413Dic6—28no—
414C3×C8—22no—
415C3×C8—41no—
416C3×C8—41no—
417C3×C8—41no—
418C3⋊C8—22no—
419C3⋊C8—22no—
420C3⋊C8—22no—
421C3⋊C8—41no—
422C24C2×C4×D311yescharacteristic
423C24C2×C4×D311yescharacteristic
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8—62no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—41no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4C2×C4×D311yescharacteristic
451C22×C4C2×C4×D311yescharacteristic
452C22×C4C2×C4×D311yescharacteristic
453C22×C4C2×C4×D311yescharacteristic
454C22×C4C23×D311yescharacteristic
455M16—124no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—32no—
468C2×C8—32no—
469C2×C8—32no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—124no—
480C2×C8#23712yes—
481C4⋊C4—22no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—62no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C22⋊C4—121no—
494C22⋊C4—121no—
495C42—61no—
496C42—61no—
497C42—61no—
498C42—61no—
499C42—61no—
500C42—61no—
501C42—61no—
502C42—61no—
503C42D3×D412yes—
504C42#23712yes—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3C2×C4×D411yescharacteristic
520C22×C3C4×#4111yescharacteristic
521C22×C3#52011yescharacteristic
522C22×C3C2×#13111yescharacteristic
523C22×C3C2×#13111yescharacteristic
524C22×C3C2×#13111yescharacteristic
525C22×C3#56811yescharacteristic
526C3×C4—22no—
527C3×C4—22no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4#54611yescharacteristic
531C3×C4#54811yescharacteristic
532C3×C4#55611yescharacteristic
533C3×C4#55911yescharacteristic
534Dic3—22no—
535Dic3—22no—
536Dic3—41no—
537Dic3—41no—
538Dic3—41no—
539Dic3—41no—
540Dic3—41no—
541Dic3—41no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23#90112yes—
556C23C22×C4×D311yescharacteristic
557Q8—68no—
558D4—44no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4#67⋊C412yes—
602C2×C4#67⋊C412yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×#23711yescharacteristic
605C2×C4C2×#23711yescharacteristic
606C2×C4#100411yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C2×C4#101011yescharacteristic
609C8—22no—
610C8—41no—
611C8—41no—
612C8—41no—
613C8—62no—
614C8—62no—
615C8—62no—
616C8—121no—
617C2×C3—42no—
618C2×C3—81no—
619C2×C3#236711yescharacteristic
620C2×C3#292911yescharacteristic
621C2×C3#293311yescharacteristic
622C2×C3#296611yescharacteristic
623C2×C3#296911yescharacteristic
624C2×C3#299311yescharacteristic
625C2×C3#299411yescharacteristic
626C22—22no—
627C22—22no—
628C22—22no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22C2×#67⋊C411yescharacteristic
641C22C4×#23711yescharacteristic
642C22#554611yescharacteristic
643C22C2×#90111yescharacteristic
644C22#574511yescharacteristic
645C22D3×#13111yescharacteristic
646C22D3×#13111yescharacteristic
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—62no—
652C4—62no—
653C4—121no—
654C4—121no—
655C4—121no—
656C4—121no—
657C4—121no—
658C4—121no—
659C4#559811yescharacteristic
660C4#561211yescharacteristic
661C4#567111yescharacteristic
662C4#571311yescharacteristic
663C3#2321011yescharacteristic, complemented, Sylow
664C2—42no—
665C2—81no—
666C2#7630711yescharacteristic
667C2#7859611yescharacteristic
668C2#7865511yescharacteristic
669C2#7940011yescharacteristic
670C2#7940911yescharacteristic
671C2#7952011yescharacteristic
672C2#7952511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1954C2
prime radicalC22×C3×D4C2×C4
cosocleC22×C3×C4C24
hypercenter#1954D3
hypocenterC3#23210
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1954

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C8×Dic6C2semidirect
5. #70473C2semidirect
6. #70442C2semidirect
7. #69951C2semidirect
8. #69950C2semidirect
9. #69949C2semidirect
10. #69947C2semidirect
11. #69914C2semidirect
12. #69913C2semidirect
13. #69913C2semidirect
14. #69912C2semidirect
15. #69755C2semidirect
663. C3#23210semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#823482
C2×#50112
C2×C8×Dic62
C3×#19542
#704732
#704422
#699512
#699502
#699492
#699472
#699142
#699132
#699122
#697552
#232103