Finite Group Databasebeta

← #10845472

Subgroups — #10845472

Order 768.

860 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#477C211yesmaximal, characteristic, complemented
2#83240C211yesmaximal, characteristic
3#82499C211yesmaximal, characteristic
4C2×#4836C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71631C211yesmaximal, characteristic, complemented
7#70637C211yesmaximal, characteristic, complemented
8#70636C211yesmaximal, characteristic, complemented
9#70631C211yesmaximal, characteristic, complemented
10#70630C211yesmaximal, characteristic, complemented
11#70455C211yesmaximal, characteristic, complemented
12#70271C211yesmaximal, characteristic, complemented
13#70270C211yesmaximal, characteristic, complemented
14#70244C211yesmaximal, characteristic, complemented
15#70243C211yesmaximal, characteristic, complemented
16#25352—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947—21no—
20C2×#946C2211yescharacteristic
21C2×#946C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C22×D3×C8C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#477—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#107C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31#5229—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#817C2211yescharacteristic
38#4992—21no—
39#4992—21no—
40#4991—21no—
41#4987C2211yescharacteristic
42#4987C2211yescharacteristic
43#4987C2211yescharacteristic
44#4987C2211yescharacteristic
45#4987C2211yescharacteristic
46#4987C2211yescharacteristic
47#4981—21no—
48#4940—21no—
49#4940—21no—
50#4936C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C2×#803C2211yescharacteristic
53#4836—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#477D311yescharacteristic, complemented
63C2×#435—31no—
64#2547—31no—
65#2536—31no—
66#2076—31no—
67#2075—31no—
68#2036—31no—
69#2035—31no—
70#1993—31no—
71#1949—31no—
72#1894—31no—
73#1892—31no—
74#1809—31no—
75#1808—31no—
76#1763—31no—
77C22×C3×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×D12D411yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×C4×D3C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95#947—44no—
96#946—22no—
97#946—22no—
98#946—41no—
99#946—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4D411yescharacteristic
109C2×(C2×D3)⋊C4D411yescharacteristic
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×ΓL(1,𝔽25)—22no—
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)—41no—
123C2×D3×C8—22no—
124C2×D3×C8—22no—
125C2×D3×C8—41no—
126C4×D12—22no—
127C4×D12—22no—
128C3×#107—22no—
129C3×#107—22no—
130C3×C4⋊C8D414yes—
131C3×#103—22no—
132C3×#103—22no—
133#835—22no—
134#835—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144(C2×D3)⋊C8—22no—
145(C2×D3)⋊C8—22no—
146#818—41no—
147#817—41no—
148#803—22no—
149C22×M16—31no—
150C23×C8—31no—
151C2×C4⋊Q8C2×D311yescharacteristic
152C2×#124—61no—
153C2×C22⋊Q8—31no—
154C2×C22⋊Q8—31no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—31no—
157#477—24no—
158#435—64no—
159C2×#107—31no—
160C2×#107C2×D311yescharacteristic
161C2×#107C2×D311yescharacteristic
162C2×#106—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170C2×#97—31no—
171#398—61no—
172#397—61no—
173#397—61no—
174#394—61no—
175#391—61no—
176#388—61no—
177#388—61no—
178#382—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C23×D3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—42no—
202C2×C3×Q8—42no—
203C2×C3×Q8—42no—
204C2×C3×Q8#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×D12—22no—
219C2×D12—22no—
220C2×D12—41no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C2×C3×C8C2×D412yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C42—42no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260C4⋊Dic3—22no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—42no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8#4112yes—
278ΓL(1,𝔽25)—44no—
279D3×C8—44no—
280C22×Q8—21no—
281C22×Q8C22×D311yescharacteristic
282C22×D4—31no—
283C23×C4—31no—
284C23×C4—31no—
285C2×M16—62no—
286C2×M16—62no—
287C2×M16—121no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8—121no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297#124—124no—
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—121no—
301C22⋊Q8—121no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×D4—62no—
305C4×D4—62no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—31no—
319C2×C22⋊C4—31no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C42—21no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#107—22no—
332#107—22no—
333#107—121no—
334#106—121no—
335C4⋊C8#6714yes—
336#103—22no—
337#103—22no—
338#103—62no—
339#103—62no—
340C22⋊C8—62no—
341C22⋊C8—62no—
342#97—62no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3C22×D411yescharacteristic
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3AGL(1,ℤ/8)12yes—
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—41no—
370C3×Q8—41no—
371C3×Q8—84no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4#12112yes—
389C2×C3×C4#12112yes—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C2×C3×C42−1+411yescharacteristic
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414D12—44no—
415C4×D3—41no—
416C4×D3—41no—
417C4×D3—41no—
418C4×D3—41no—
419C4×D3—41no—
420C4×D3—41no—
421C4×D3—41no—
422C4×D3—41no—
423C3×C8—22no—
424C3×C8—22no—
425C3⋊C8—22no—
426C3⋊C8—41no—
427C24—31no—
428C24—31no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8#23912yes—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—121no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466M16—124no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—32no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—124no—
478C2×C8C2×#6712yes—
479C2×C8C2×#6712yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—62no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C22⋊C4—62no—
500C22⋊C4—62no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C42—42no—
510C42—122no—
511C42#23712yes—
512C42C2×#6712yes—
513C22×C3C2×#12111yescharacteristic
514C22×C3#54611yescharacteristic
515C22×C3#54711yescharacteristic
516C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
517C22×C3#57511yescharacteristic
518C22×C3#57811yescharacteristic
519C22×C3#57811yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—41no—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—42no—
537C3×C4—81no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4#52911yescharacteristic
541C3×C4#54211yescharacteristic
542C3×C4#54611yescharacteristic
543C3×C4#54811yescharacteristic
544Dic3—41no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548Dic3—81no—
549Dic3—81no—
550C23—32no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23C2×D3×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—84no—
570D4—124no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4#93812yes—
614C2×C4#93812yes—
615C2×C4C2×#23711yescharacteristic
616C2×C4C2×#23911yescharacteristic
617C2×C4#100711yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C2×C4#100911yescharacteristic
620C2×C4C22×#6711yescharacteristic
621C8—22no—
622C8—22no—
623C8—62no—
624C8—121no—
625C2×C3#242711yescharacteristic
626C2×C3#305511yescharacteristic
627C2×C3#305911yescharacteristic
628C2×C3#330811yescharacteristic
629C2×C3#331311yescharacteristic
630C2×C3#334311yescharacteristic
631C2×C3#334911yescharacteristic
632D3—42no—
633D3—81no—
634C22—62no—
635C22—62no—
636C22—62no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22—121no—
645C22—121no—
646C22—121no—
647C22—121no—
648C22#563011yescharacteristic
649C22#563111yescharacteristic
650C22#576611yescharacteristic
651C22#576611yescharacteristic
652C22D3×AGL(1,ℤ/8)11yescharacteristic
653C22#577311yescharacteristic
654C22C2×#93811yescharacteristic
655C4—22no—
656C4—22no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—121no—
662C4—121no—
663C4—241no—
664C4—241no—
665C4—241no—
666C4—241no—
667C4#562411yescharacteristic
668C4#564311yescharacteristic
669C4#581711yescharacteristic
670C4#582911yescharacteristic
671C3#2535211yescharacteristic, complemented, Sylow
672C2—122no—
673C2—241no—
674C2#8112111yescharacteristic
675C2#8112211yescharacteristic
676C2#8112711yescharacteristic
677C2#8112811yescharacteristic
678C2#8275611yescharacteristic
679C2#8388711yescharacteristic
680C2#8395011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenterC2×#477D3
hypocenterC3#25352
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×C3×#477C2semidirect
5. #71788C2semidirect
6. #71631C2semidirect
7. #70637C2semidirect
8. #70636C2semidirect
9. #70631C2semidirect
10. #70630C2semidirect
11. #70455C2semidirect
12. #70271C2semidirect
13. #70270C2semidirect
14. #70244C2semidirect
15. #70243C2semidirect
62. C2×#477D3semidirect
671. C3#25352semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4772
#832402
#824992
C2×#48362
#717882
#716312
#706372
#706362
#706312
#706302
#704552
#702712
#702702
#702442
#702432
#253523