Finite Group Databasebeta

← #10818158

Subgroups — #10818158

Order 768.

824 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#924C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic
3C2×#4701C211yesmaximal, characteristic, complemented
4C3×#1749C211yesmaximal, characteristic
5#69915C211yesmaximal, characteristic
6#69914C211yesmaximal, characteristic
7#69913C211yesmaximal, characteristic
8#69912C211yesmaximal, characteristic
9#69838C211yesmaximal, characteristic, complemented
10#69836C211yesmaximal, characteristic, complemented
11#69810C211yesmaximal, characteristic, complemented
12#68655C211yesmaximal, characteristic, complemented
13#68655C211yesmaximal, characteristic, complemented
14#68653C211yesmaximal, characteristic, complemented
15#68651C211yesmaximal, characteristic, complemented
16#23052—31nomaximal, Sylow
17C22×C3×C42C2211yescharacteristic
18C2×#924C411yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C2×C3×C22⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24#5212C411yescharacteristic
25#5210C411yescharacteristic
26#5210C411yescharacteristic
27C4×#217C2211yescharacteristic
28C4×#217C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#815C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4933C411yescharacteristic
41#4933C411yescharacteristic
42C4×Dic3⋊C4C2211yescharacteristic
43C4×Dic3⋊C4C2211yescharacteristic
44#4929C411yescharacteristic
45#4928C411yescharacteristic
46#4925—24no—
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50C2×C4×C3⋊C8C2211yescharacteristic
51#4701—24no—
52C3×#336C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#446—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65#1985—31no—
66#1940—31no—
67#1924—31no—
68#1924—31no—
69#1923—31no—
70#1923—31no—
71#1916—31no—
72#1916—31no—
73#1771—31no—
74#1771—31no—
75C8×C4⋊C4—31no—
76#1749D311yescharacteristic
77C23×C3×C4C2×C411yescharacteristic
78C23×C3×C4C2×C411yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2×C411yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C42—22no—
86C2×C3×C42—22no—
87C2×C3×C42C2×C411yescharacteristic
88C2×C3×C42C2×C411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217C2×C411yescharacteristic
92C2×#217C2×C411yescharacteristic
93C2×#217C2×C411yescharacteristic
94C2×#217C2×C411yescharacteristic
95C2×C4⋊Dic3C2×C411yescharacteristic
96C2×C4⋊Dic3C2×C411yescharacteristic
97C2×C4⋊Dic3C2×C411yescharacteristic
98#924—24no—
99C2×Dic3⋊C4C2×C411yescharacteristic
100C2×Dic3⋊C4C2×C411yescharacteristic
101C2×Dic3⋊C4C2×C411yescharacteristic
102C2×Dic3⋊C4C2×C411yescharacteristic
103C2×Dic3⋊C4C2×C411yescharacteristic
104C2×Dic3⋊C4C2×C411yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C4×Dic6—22no—
114C4×Dic6—22no—
115C3×C22⋊C8—22no—
116C3×C22⋊C8—22no—
117C3×#97—22no—
118C3×#97—22no—
119C3⋊#95C2×C411yescharacteristic
120C3⋊#95C2×C411yescharacteristic
121C3⋊#95C2×C411yescharacteristic
122C3⋊#95C2×C411yescharacteristic
123C3⋊#95C2×C411yescharacteristic
124C3⋊#95C2×C411yescharacteristic
125C3⋊#95C2×C411yescharacteristic
126C3⋊#95C2×C411yescharacteristic
127C3⋊#95C2×C411yescharacteristic
128C3⋊#95C2×C411yescharacteristic
129#830—22no—
130#830—22no—
131#815—22no—
132#815—22no—
133Dic3⋊C8—22no—
134Dic3⋊C8—22no—
135Dic3⋊C8—22no—
136Dic3⋊C8—22no—
137C8×Dic3—22no—
138C8×Dic3—22no—
139C4⋊(C3⋊C8)—22no—
140C4⋊(C3⋊C8)D414yes—
141C4×C3⋊C8—22no—
142C4×C3⋊C8—22no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C22×C42C2×D311yescharacteristic
146#446—64no—
147#431—64no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×C4⋊C8—31no—
153C2×C4⋊C8—31no—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8C2×D311yescharacteristic
157C2×C22⋊C8C2×D311yescharacteristic
158C2×#97—31no—
159C2×#97—31no—
160C2×#97C2×D311yescharacteristic
161C2×#97C2×D311yescharacteristic
162C2×C4×C8—31no—
163C2×C4×C8—31no—
164C2×C4×C8—31no—
165C2×C4×C8—31no—
166#391—31no—
167#389—31no—
168#387—31no—
169#387—31no—
170#385—31no—
171#385—31no—
172#380—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C22⋊C4—31no—
177C4×C22⋊C4—31no—
178#336—31no—
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185#336—31no—
186#336C2×D311yescharacteristic
187#336C2×D311yescharacteristic
188C24×C3C22×C411yescharacteristic
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—41no—
201C22×C3×C4C22×C411yescharacteristic
202C22×C3×C4C22×C411yescharacteristic
203C22×C3×C4C22×C411yescharacteristic
204C22×C3×C4C22×C411yescharacteristic
205C22×C3×C4C22×C411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3C22×C411yescharacteristic
208C22×Dic3C22×C411yescharacteristic
209C22×Dic3C22×C411yescharacteristic
210C22×Dic3C22×C411yescharacteristic
211C22×Dic3C22×C411yescharacteristic
212C22×Dic3C22×C411yescharacteristic
213C22×Dic3C22×C411yescharacteristic
214C22×Dic3C22×C411yescharacteristic
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—41no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8#4112yes—
229C2×C3×C8#4112yes—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42C2×D412yes—
235C3×C42#4112yes—
236#217—22no—
237#217—22no—
238#217—22no—
239#217—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3—41no—
243Dic3⋊C4—22no—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4M1614yes—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C2×D412yes—
268C2×C3⋊C8C2×D412yes—
269C22×Q8—31no—
270C23×C4C4×D311yescharacteristic
271C23×C4C4×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C22⋊Q8—64no—
282C4×Q8—62no—
283C4×Q8—62no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42C4×D311yescharacteristic
304C2×C42C4×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D311yescharacteristic
307C4⋊C8—34no—
308C4⋊C8—62no—
309C4⋊C8—62no—
310C4⋊C8—62no—
311C4⋊C8—62no—
312C4⋊C8—62no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—62no—
316C22⋊C8—62no—
317#97—22no—
318#97—22no—
319#97—62no—
320#97—62no—
321C4×C8—62no—
322C4×C8—62no—
323C4×C8—62no—
324C4×C8—62no—
325#95—31no—
326#95—31no—
327#95—31no—
328#95—31no—
329#95—31no—
330#95—31no—
331#95—31no—
332#95—31no—
333#95—31no—
334#95—31no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3#13112yes—
342C23×C3C23×C411yescharacteristic
343C2×C3×C4—28no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—44no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4C4×D412yes—
361C2×C3×C4C4×D412yes—
362C2×C3×C4C2×M1612yes—
363C2×C3×C4#13112yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C42−1+411yescharacteristic
369C2×C3×C42−1+411yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3C2×M1612yes—
393C2×Dic3C2×M1612yes—
394Dic6—44no—
395C3×C8—22no—
396C3×C8—22no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—22no—
400C3⋊C8—22no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C24C2×C4×D311yescharacteristic
404C2×Q8—62no—
405C2×Q8—62no—
406C2×Q8—121no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—41no—
427C22×C4C2×C4×D311yescharacteristic
428C22×C4C2×C4×D311yescharacteristic
429C22×C4C2×C4×D311yescharacteristic
430C22×C4C2×C4×D311yescharacteristic
431C22×C4C2×C4×D311yescharacteristic
432C22×C4C23×D311yescharacteristic
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—32no—
444C2×C8—32no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8#23512yes—
456C2×C8#23512yes—
457C4⋊C4—34no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—121no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C42—22no—
471C42—22no—
472C42—21no—
473C42—21no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—61no—
482C42D3×D412yes—
483C42#23712yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3C2×C4×D411yescharacteristic
492C22×C3C4×#4111yescharacteristic
493C22×C3#51911yescharacteristic
494C22×C3C22×M1611yescharacteristic
495C22×C3C2×#13111yescharacteristic
496C22×C3C2×#13111yescharacteristic
497C22×C3#56811yescharacteristic
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4#54711yescharacteristic
504C3×C4#54811yescharacteristic
505C3×C4#55211yescharacteristic
506C3×C4#55611yescharacteristic
507Dic3—22no—
508Dic3—22no—
509Dic3—41no—
510Dic3—41no—
511Dic3—41no—
512Dic3—41no—
513Dic3—41no—
514Dic3—41no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#90712yes—
522C23C22×C4×D311yescharacteristic
523Q8—124no—
524C2×C4—28no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4#67⋊C412yes—
566C2×C4#67⋊C412yes—
567C2×C4D3×M1612yes—
568C2×C4#90712yes—
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23511yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×#23711yescharacteristic
573C2×C4#101011yescharacteristic
574C2×C4#101011yescharacteristic
575C8—22no—
576C8—22no—
577C8—41no—
578C8—41no—
579C8—62no—
580C8—62no—
581C8—121no—
582C8—121no—
583C2×C3—42no—
584C2×C3#236211yescharacteristic
585C2×C3#292911yescharacteristic
586C2×C3D4×M1611yescharacteristic
587C2×C3#296511yescharacteristic
588C2×C3#296811yescharacteristic
589C2×C3#298411yescharacteristic
590C2×C3#298511yescharacteristic
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22C4×#23511yescharacteristic
599C22C2×#67⋊C411yescharacteristic
600C22#551611yescharacteristic
601C22C2×D3×M1611yescharacteristic
602C22C2×#90711yescharacteristic
603C22C2×#90711yescharacteristic
604C22#574511yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—62no—
611C4—62no—
612C4—121no—
613C4—121no—
614C4—121no—
615C4—121no—
616C4—121no—
617C4—121no—
618C4#554111yescharacteristic
619C4#554211yescharacteristic
620C4#555111yescharacteristic
621C4#555211yescharacteristic
622C3#2305211yescharacteristic, complemented, Sylow
623C2—42no—
624C2#7346811yescharacteristic
625C2#7848111yescharacteristic
626C2#7852311yescharacteristic
627C2#7862611yescharacteristic
628C2#7862811yescharacteristic
629C2#7931511yescharacteristic
630C2#7931611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1749C2
prime radicalC24×C3C22×C4
cosocleC22×C3×C4C24
hypercenter#1749D3
hypocenterC3#23052
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. C2×#4701C2semidirect
9. #69838C2semidirect
10. #69836C2semidirect
11. #69810C2semidirect
12. #68655C2semidirect
13. #68655C2semidirect
14. #68653C2semidirect
15. #68651C2semidirect
622. C3#23052semidirect

Maximal subgroups

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

subgroupindex
C4×#9242
C2×#49252
C2×#47012
C3×#17492
#699152
#699142
#699132
#699122
#698382
#698362
#698102
#686552
#686532
#686512
#230523