Finite Group Databasebeta

← #10784423

Subgroups — #10784423

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3228C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic, complemented
3#83325C211yesmaximal, characteristic, complemented
4#83318C211yesmaximal, characteristic
5#83270C211yesmaximal, characteristic
6#79272C211yesmaximal, characteristic, complemented
7#71368C211yesmaximal, characteristic, complemented
8#71362C211yesmaximal, characteristic, complemented
9#71341C211yesmaximal, characteristic, complemented
10#71339C211yesmaximal, characteristic, complemented
11#71195C211yesmaximal, characteristic, complemented
12#71192C211yesmaximal, characteristic, complemented
13#71190C211yesmaximal, characteristic, complemented
14#71183C211yesmaximal, characteristic, complemented
15#70744C211yesmaximal, characteristic, complemented
16#19744—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19#5470C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#498C2211yescharacteristic
23C3×#486C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#482—22no—
26C3×#478—22no—
27C3×#431C2211yescharacteristic
28#5239—22no—
29#5093C2211yescharacteristic
30#5093C2211yescharacteristic
31#5092C2211yescharacteristic
32#5092C2211yescharacteristic
33#5092C2211yescharacteristic
34#5080C2211yescharacteristic
35#5079C2211yescharacteristic
36#5079C2211yescharacteristic
37#5079C2211yescharacteristic
38#5078C2211yescharacteristic
39#5078C2211yescharacteristic
40#5078C2211yescharacteristic
41#5076C2211yescharacteristic
42#5070C2211yescharacteristic
43#5070C2211yescharacteristic
44#5070C2211yescharacteristic
45#5054—22no—
46#5050—22no—
47#5045—22no—
48#5033—22no—
49#5031—22no—
50#5022—22no—
51#5007—22no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54#4850—22no—
55#4722—22no—
56#4682C2211yescharacteristic
57#4682C2211yescharacteristic
58#4681C2211yescharacteristic
59#4681C2211yescharacteristic
60#4455C2211yescharacteristic
61#4454C2211yescharacteristic
62#4454C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3231—31no—
65#3230—31no—
66#3228D311yescharacteristic
67#3226—31no—
68#3210—31no—
69#3209—31no—
70#1723—31no—
71#1723—31no—
72#1722—31no—
73#1720—31no—
74#1719—31no—
75#1718—31no—
76#1713—31no—
77#1711—31no—
78#1454—31no—
79C2×C3×#41—21no—
80C22×Dic6—21no—
81C3×C4⋊Q8C2311yescharacteristic
82C3×C4⋊Q8C2311yescharacteristic
83C3×C4⋊Q8C2311yescharacteristic
84C3×#127C2311yescharacteristic
85C3×#125—21no—
86C3×#124—21no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—21no—
89C3×#121—21no—
90C3×#121—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×#117—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#216—22no—
98C2×#216—22no—
99C2×#216—22no—
100C2×#214—22no—
101#926—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104#924—21no—
105#924—21no—
106#870—21no—
107#869—21no—
108#869—21no—
109#869—21no—
110#869C2311yescharacteristic
111#869C2311yescharacteristic
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114C4×Dic6—21no—
115C4×Dic6—21no—
116C3×#107—22no—
117C3×#106—22no—
118C3×C4⋊C8D412yes—
119C3×#102—22no—
120C3×#102—22no—
121C3×C22⋊C8D412yes—
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124#835—22no—
125#835—22no—
126#835—22no—
127#832—22no—
128#832—22no—
129#832—22no—
130#830D412yes—
131#830D412yes—
132#818—22no—
133#818—22no—
134#816—22no—
135#816—22no—
136#810—22no—
137#810—22no—
138#808—22no—
139#807—22no—
140#807—22no—
141#807—22no—
142C4⋊(C3⋊C8)D412yes—
143C4⋊(C3⋊C8)D412yes—
144C4×C3⋊C8C2311yescharacteristic
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148#533—31no—
149#533C2×D311yescharacteristic
150#532C2×D311yescharacteristic
151C2×C4⋊Q8—31no—
152#500—31no—
153#500—31no—
154#500—31no—
155#500—31no—
156#500—31no—
157#500C2×D311yescharacteristic
158#498—31no—
159#498C2×D311yescharacteristic
160#494—31no—
161#494—31no—
162#494—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486C2×D311yescharacteristic
175#486C2×D311yescharacteristic
176#485—62no—
177#484—62no—
178#482—22no—
179#480—62no—
180#479—62no—
181#478—22no—
182#477—62no—
183#474—62no—
184#464—62no—
185#462—62no—
186#452—62no—
187#451—62no—
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#322—31no—
192#322—31no—
193#322—31no—
194C82—31no—
195C3×#41—82no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—41no—
203C22×C3×C4—41no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×Dic3—41no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—41no—
213C2×Dic6—82no—
214C2×C3×C8C2×D412yes—
215C2×C3×C8C2×D412yes—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4—41no—
224C3×C22⋊C4—41no—
225C3×C22⋊C4—41no—
226C3×C22⋊C4—41no—
227C3×C42—41no—
228C3×C42C2×D411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2411yescharacteristic
232#217—41no—
233#216—82no—
234#216—82no—
235#216—82no—
236#214—82no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—41no—
246C4⋊Dic3—41no—
247Dic3⋊C4—41no—
248Dic3⋊C4—41no—
249C4×Dic3—41no—
250C2×C3⋊C8C2×D412yes—
251C2×C3⋊C8C2×D412yes—
252C2×C3⋊C8C2×D412yes—
253C2×C3⋊C8C2×D412yes—
254C2×#41—21no—
255C22×Q8—61no—
256C2×Q16—62no—
257C2×Q16—62no—
258C2×Q16—62no—
259C2×SD16—62no—
260C4⋊Q8—31no—
261C4⋊Q8—31no—
262C4⋊Q8—31no—
263C4⋊Q8—31no—
264C4⋊Q8—61no—
265C4⋊Q8—61no—
266C4⋊Q8—61no—
267C4⋊Q8C22×D311yescharacteristic
268C4⋊Q8C22×D311yescharacteristic
269C4⋊Q8C22×D311yescharacteristic
270#127C22×D311yescharacteristic
271#125—21no—
272#125—61no—
273#124—21no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—61no—
277C22⋊Q8—61no—
278#121—21no—
279#121—21no—
280C4×Q8—21no—
281C4×Q8—21no—
282C4×Q8—61no—
283C4×Q8—61no—
284C4×D4—21no—
285C4×D4—21no—
286#117—21no—
287#117—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C42C22×D311yescharacteristic
291#107—22no—
292#107—62no—
293#107—62no—
294#107—62no—
295#107—62no—
296#107—62no—
297#106—22no—
298#106—62no—
299C4⋊C8—32no—
300C4⋊C8—32no—
301C4⋊C8#6712yes—
302#103—62no—
303#103—62no—
304#103—62no—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—22no—
310#102—22no—
311#102—62no—
312#102—62no—
313#102—62no—
314C22⋊C8—32no—
315C22⋊C8—32no—
316C22⋊C8#6712yes—
317C4×C8—31no—
318C4×C8—31no—
319C4×C8—31no—
320C4×C8—31no—
321C4×C8C22×D311yescharacteristic
322C4×C8C22×D311yescharacteristic
323C23×C3—41no—
324C23×C3#12711yescharacteristic
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—82no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—82no—
339C2×C3×C4#12711yescharacteristic
340C2×C3×C4#12711yescharacteristic
341C2×C3×C4#12711yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—82no—
350Dic6—42no—
351Dic6—42no—
352Dic6—42no—
353Dic6—42no—
354Dic6—82no—
355C3×C8—22no—
356C3×C8—22no—
357C3⋊C8—22no—
358C3⋊C8—22no—
359C3⋊C8—22no—
360C3⋊C8—22no—
361#41—82no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—61no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—121no—
370C2×Q8—242no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—41no—
375C22×C4—41no—
376C22×C4—121no—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D411yescharacteristic
379C22×C4C2×#6711yescharacteristic
380Q16—242no—
381Q16—242no—
382Q16—242no—
383SD16—242no—
384C2×C8—32no—
385C2×C8—32no—
386C2×C8—32no—
387C2×C8—32no—
388C2×C8C2×#6712yes—
389C2×C8C2×#6712yes—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C4⋊C4—121no—
409C4⋊C4—121no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—121no—
414C42—41no—
415C42—121no—
416C42D3×D411yescharacteristic
417C42D3×D411yescharacteristic
418C42C2×#6711yescharacteristic
419C42C23×D311yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3C2×#12711yescharacteristic
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—81no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4#57612yes—
434C3×C4#57812yes—
435C3×C4#57812yes—
436Dic3—81no—
437Dic3—81no—
438Dic3—81no—
439Dic3—81no—
440C23—41no—
441C23#94011yescharacteristic
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—122no—
446Q8—122no—
447Q8—122no—
448Q8—122no—
449Q8—242no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—82no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—242no—
466C2×C4#94011yescharacteristic
467C2×C4#94011yescharacteristic
468C2×C4#94011yescharacteristic
469C2×C4C2×D3×D411yescharacteristic
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C22×#6711yescharacteristic
472C8—22no—
473C8—22no—
474C8—62no—
475C8—62no—
476C8—62no—
477C8—62no—
478C2×C3—41no—
479C2×C3—161no—
480C2×C3#315511yescharacteristic
481C2×C3#315611yescharacteristic
482C2×C3#315611yescharacteristic
483C22—21no—
484C22—21no—
485C22—21no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22C2×#94011yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—241no—
497C4—241no—
498C4—241no—
499C4—241no—
500C4#577112yes—
501C4#577412yes—
502C4#583812yes—
503C3#1974411yescharacteristic, complemented, Sylow
504C2—41no—
505C2—161no—
506C2#8340311yescharacteristic
507C2#8390711yescharacteristic
508C2#8394011yescharacteristic
subgroupG/H
centerC22C2×#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3228C2
socleC22×C3C2×#127
prime radicalC3×#532C22
hypercenter#3228D3
hypocenterC3#19744
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3228

Complements

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

NH ≅ G/Ntype
2. #83370C2semidirect
3. #83325C2semidirect
6. #79272C2semidirect
7. #71368C2semidirect
8. #71362C2semidirect
9. #71341C2semidirect
10. #71339C2semidirect
11. #71195C2semidirect
12. #71192C2semidirect
13. #71190C2semidirect
14. #71183C2semidirect
15. #70744C2semidirect
503. C3#19744semidirect

Maximal subgroups

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

subgroupindex
C3×#32282
#833702
#833252
#833182
#832702
#792722
#713682
#713622
#713412
#713392
#711952
#711922
#711902
#711832
#707442
#197443