Finite Group Databasebeta

← #10898410

Subgroups — #10898410

Order 768.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73430C211yesmaximal, characteristic
2#73386C211yesmaximal, characteristic, complemented
3C3×#1768C211yesmaximal, characteristic, complemented
4#70488C211yesmaximal, characteristic, complemented
5#70483C211yesmaximal, characteristic
6#70482C211yesmaximal, characteristic
7#70464C211yesmaximal, characteristic, complemented
8#69818C211yesmaximal, characteristic
9#69817C211yesmaximal, characteristic, complemented
10#69774C211yesmaximal, characteristic
11#69769C211yesmaximal, characteristic, complemented
12#68686C211yesmaximal, characteristic
13#68684C211yesmaximal, characteristic, complemented
14#68626C211yesmaximal, characteristic
15#68623C211yesmaximal, characteristic, complemented
16#29442—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#376C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#817C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33#4939C2211yescharacteristic
34#4939C2211yescharacteristic
35#4939C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37#4929C2211yescharacteristic
38#4929C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4676—21no—
45#4675—21no—
46#4675—21no—
47#4675—21no—
48#4652—21no—
49#4652—21no—
50#4651—21no—
51#4650—21no—
52C3×#336C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2644—31no—
61#2320—31no—
62#2088—31no—
63#2066—31no—
64#2065—31no—
65#2055—31no—
66#1968—31no—
67#1941—31no—
68#1936—31no—
69#1935—31no—
70#1926—31no—
71#1920—31no—
72#1880—31no—
73#1865—31no—
74#1768D311yescharacteristic, complemented
75C22×D12C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×#204—22no—
105C2×#204—22no—
106C2×#204—41no—
107C4×Dic6—22no—
108C4×Dic6—22no—
109C3×C4⋊C8—22no—
110C3×C4⋊C8—22no—
111C3×#97—41no—
112C3×C4×C8—41no—
113C3×#95D411yescharacteristic
114C3×#95D411yescharacteristic
115C3×#95D411yescharacteristic
116C3×#95D411yescharacteristic
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127#821—22no—
128#821—22no—
129#821—41no—
130#817—22no—
131#817—22no—
132#817—41no—
133#816—22no—
134#816—22no—
135#816—41no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×#106—31no—
139C2×#106—31no—
140C2×#106—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×C4⋊C8C2×D311yescharacteristic
143C2×#103—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#97C2×D311yescharacteristic
150C2×C4×C8C2×D311yescharacteristic
151#397—61no—
152#397—61no—
153#397—61no—
154#396—61no—
155#391—31no—
156#391—31no—
157#390—31no—
158#390—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389—31no—
164#389—31no—
165#387—61no—
166#387—61no—
167#383—61no—
168#382—61no—
169C4×C4⋊C4—31no—
170#376C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6#4112yes—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8#4112yes—
217C2×C3×C8#4112yes—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224(C2×D3)⋊C4—81no—
225(C2×D3)⋊C4—81no—
226(C2×D3)⋊C4—81no—
227(C2×D3)⋊C4—81no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3#4112yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—81no—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—24no—
249C4×Dic3—81no—
250#204—44no—
251C22×Q8—31no—
252C22×D4—31no—
253C2×SD16—62no—
254C2×SD16—62no—
255C2×SD16—121no—
256C22×C8C22×D311yescharacteristic
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260C4×Q8—62no—
261C4×Q8—62no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—61no—
281C2×C42—61no—
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284C2×C42C22×D311yescharacteristic
285#106—62no—
286#106—62no—
287#106—121no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290#103—62no—
291#103—62no—
292#103—121no—
293#102—62no—
294#102—62no—
295#102—121no—
296#97—41no—
297C4×C8—41no—
298#95—61no—
299#95—61no—
300#95—61no—
301#95—61no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95D1211yescharacteristic
309#95D1211yescharacteristic
310#95D1211yescharacteristic
311#95D1211yescharacteristic
312C23×C3C22×D411yescharacteristic
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365D12—41no—
366D12—41no—
367D12—41no—
368D12—41no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3×C8—22no—
374C3×C8—22no—
375C3×C8—41no—
376C3×C8—41no—
377C24—61no—
378C2×Q8—32no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4C2×D1211yescharacteristic
398C22×C4C2×D1211yescharacteristic
399C22×C4C2×D1211yescharacteristic
400C22×C4C2×D1211yescharacteristic
401C22×C4C2×D1211yescharacteristic
402C22×C4C2×D1211yescharacteristic
403C22×C4C23×D311yescharacteristic
404SD16—124no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8#23512yes—
416C2×C8#23512yes—
417C4⋊C4—32no—
418C4⋊C4—64no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—241no—
431C4⋊C4—241no—
432C4⋊C4—241no—
433C4⋊C4—241no—
434C4⋊C4—241no—
435C22⋊C4—241no—
436C22⋊C4—241no—
437C22⋊C4—241no—
438C22⋊C4—241no—
439C22⋊C4—241no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—64no—
448C42—241no—
449C22×C3#52511yescharacteristic
450C22×C3#53211yescharacteristic
451C22×C3#53711yescharacteristic
452C22×C3#57711yescharacteristic
453C22×C3#57711yescharacteristic
454C22×C3#57711yescharacteristic
455C22×C3#57711yescharacteristic
456C2×D3—81no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—81no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4#55111yescharacteristic
470C3×C4#55211yescharacteristic
471C3×C4#55311yescharacteristic
472C3×C4#55611yescharacteristic
473Dic3—42no—
474Dic3—42no—
475Dic3—81no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23C22×D1211yescharacteristic
488Q8—62no—
489Q8—62no—
490Q8—121no—
491Q8—121no—
492D4—121no—
493D4—121no—
494D4—121no—
495D4—121no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4C2×#23511yescharacteristic
536C2×C4C2×#23511yescharacteristic
537C2×C4#100411yescharacteristic
538C2×C4#100711yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4D3×#4111yescharacteristic
541C8—22no—
542C8—22no—
543C8—41no—
544C8—41no—
545C2×C3#260111yescharacteristic
546C2×C3#312511yescharacteristic
547C2×C3#312611yescharacteristic
548C2×C3#317311yescharacteristic
549C2×C3#317411yescharacteristic
550C2×C3#326011yescharacteristic
551C2×C3#326311yescharacteristic
552D3—161no—
553C22—241no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22—241no—
560C22#547711yescharacteristic
561C22#551011yescharacteristic
562C22#552411yescharacteristic
563C22#575111yescharacteristic
564C22#575111yescharacteristic
565C22#575111yescharacteristic
566C22#575111yescharacteristic
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—122no—
574C4—122no—
575C4—241no—
576C4—241no—
577C4—241no—
578C4—241no—
579C4—241no—
580C4#554011yescharacteristic
581C4#555611yescharacteristic
582C4#556811yescharacteristic
583C4#558111yescharacteristic
584C3#2944211yescharacteristic, complemented, Sylow
585C2—481no—
586C2#7381311yescharacteristic
587C2#7856611yescharacteristic
588C2#7856711yescharacteristic
589C2#7909611yescharacteristic
590C2#7909811yescharacteristic
591C2#7938911yescharacteristic
592C2#7939511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1768C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1768D3
hypocenterC3#29442
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
2. #73386C2semidirect
3. C3×#1768C2semidirect
4. #70488C2semidirect
7. #70464C2semidirect
9. #69817C2semidirect
11. #69769C2semidirect
13. #68684C2semidirect
15. #68623C2semidirect
74. #1768D3semidirect
584. C3#29442semidirect

Maximal subgroups

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

subgroupindex
#734302
#733862
C3×#17682
#704882
#704832
#704822
#704642
#698182
#698172
#697742
#697692
#686862
#686842
#686262
#686232
#294423