Finite Group Databasebeta

← #10784382

Subgroups — #10784382

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3226C211yesmaximal, characteristic, complemented
2#83303C211yesmaximal, characteristic
3#83272C211yesmaximal, characteristic, complemented
4#83222C211yesmaximal, characteristic, complemented
5#83216C211yesmaximal, characteristic, complemented
6#79306C211yesmaximal, characteristic, complemented
7#71368C211yesmaximal, characteristic, complemented
8#71366C211yesmaximal, characteristic, complemented
9#71363C211yesmaximal, characteristic, complemented
10#71360C211yesmaximal, characteristic, complemented
11#71193C211yesmaximal, characteristic, complemented
12#71192C211yesmaximal, characteristic, complemented
13#71186C211yesmaximal, characteristic, complemented
14#71182C211yesmaximal, characteristic, complemented
15#70744C211yesmaximal, characteristic, complemented
16#19741—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19#5477C2211yescharacteristic
20#5470C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#500C2211yescharacteristic
23C3×#486C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#480—22no—
26C3×#474—22no—
27C3×#431C2211yescharacteristic
28#5238—22no—
29#5228—22no—
30#5092C2211yescharacteristic
31#5092C2211yescharacteristic
32#5090—22no—
33#5088C2211yescharacteristic
34#5088C2211yescharacteristic
35#5085C2211yescharacteristic
36#5084—22no—
37#5080C2211yescharacteristic
38#5080C2211yescharacteristic
39#5080C2211yescharacteristic
40#5079C2211yescharacteristic
41#5079C2211yescharacteristic
42#5079C2211yescharacteristic
43#5078C2211yescharacteristic
44#5073C2211yescharacteristic
45#5070C2211yescharacteristic
46#5070C2211yescharacteristic
47#5070C2211yescharacteristic
48#5057—22no—
49#5035—22no—
50#5009—22no—
51#4968—22no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54#4845—22no—
55#4733—22no—
56#4699C2211yescharacteristic
57#4694C2211yescharacteristic
58#4693C2211yescharacteristic
59#4681C2211yescharacteristic
60#4455C2211yescharacteristic
61#4454C2211yescharacteristic
62#4454C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3228—31no—
65#3227—31no—
66#3226D311yescharacteristic, complemented
67#3210—31no—
68#3209—31no—
69#3207—31no—
70#1723—31no—
71#1722—31no—
72#1720—31no—
73#1720—31no—
74#1717—31no—
75#1714—31no—
76#1709—31no—
77#1707—31no—
78#1454—31no—
79C2×C3×#41—21no—
80C22×C3×Q8—21no—
81C2×#235—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×#127C2311yescharacteristic
89C3×#124—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#216—22no—
98C2×#216—22no—
99C2×#215—22no—
100C2×#213—22no—
101#930—21no—
102#930—21no—
103#926—21no—
104#924—21no—
105#924—21no—
106C2×Dic12—22no—
107C2×#204—22no—
108#875—21no—
109#874C2311yescharacteristic
110C4×D12—21no—
111C4×D12—21no—
112#870—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6—21no—
117C4×Dic6—21no—
118C3×#107—22no—
119C3×#107—22no—
120C3×C4⋊C8D412yes—
121C3×#102—22no—
122C3×#102—22no—
123C3×C22⋊C8D412yes—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126#835—22no—
127#835—22no—
128#835—22no—
129#832—22no—
130#832—22no—
131#832—22no—
132#830D412yes—
133#830D412yes—
134#821—22no—
135#816—22no—
136#809—22no—
137#809—22no—
138#808—22no—
139#808—22no—
140#808—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#532—31no—
150#532C2×D311yescharacteristic
151C2×C4⋊Q8C2×D311yescharacteristic
152#500—31no—
153#500C2×D311yescharacteristic
154#500C2×D311yescharacteristic
155#498—31no—
156#498—31no—
157#498—31no—
158#494—31no—
159#494—31no—
160#494—31no—
161#493—31no—
162#492—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#486—31no—
169#486—31no—
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486C2×D311yescharacteristic
175#486C2×D311yescharacteristic
176#482—62no—
177#481—62no—
178#480—22no—
179#478—62no—
180#476—62no—
181#474—22no—
182#464—62no—
183#463—62no—
184#462—62no—
185#452—62no—
186#451—62no—
187#449—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×Q8—41no—
200C2×C3×Q8—82no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—41no—
205C22×C3×C4—41no—
206C22×C3×C4—41no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C2×#67—41no—
211#235—82no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×C4×D3—41no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×C3×C8C2×D412yes—
220C2×C3×C8C2×D412yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4—41no—
229C3×C22⋊C4—41no—
230C3×C22⋊C4—41no—
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2411yescharacteristic
235#217—41no—
236#216—82no—
237#216—82no—
238#215—82no—
239#213—82no—
240(C2×D3)⋊C4—41no—
241(C2×D3)⋊C4—41no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248Dic3⋊C4—41no—
249Dic3⋊C4—41no—
250C4×Dic3—41no—
251C2×C3⋊C8C2×D412yes—
252C2×C3⋊C8C2×D412yes—
253C2×C3⋊C8C2×D412yes—
254C2×C3⋊C8C2×D412yes—
255Dic12—82no—
256#204—82no—
257C2×#41—21no—
258C2×#41—61no—
259C22×Q8—21no—
260C2×Q16—62no—
261C2×Q16—62no—
262C2×Q16—62no—
263C2×SD16—62no—
264C2×SD16—62no—
265C2×D8—62no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—31no—
270C4⋊Q8—31no—
271C4⋊Q8—31no—
272C4⋊Q8C22×D311yescharacteristic
273C4⋊Q8C22×D311yescharacteristic
274C4⋊Q8C22×D311yescharacteristic
275#127—31no—
276#127C22×D311yescharacteristic
277#125—61no—
278#124—21no—
279#124—61no—
280C22⋊Q8—61no—
281C22⋊Q8—61no—
282#121—21no—
283#121—21no—
284#121—61no—
285#121—61no—
286C4×Q8—61no—
287C4×Q8—61no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—61no—
291C4×D4—61no—
292#117—61no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C42C22×D311yescharacteristic
296#107—22no—
297#107—22no—
298#107—62no—
299#106—62no—
300#106—62no—
301#106—62no—
302C4⋊C8—32no—
303C4⋊C8—32no—
304C4⋊C8#6712yes—
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—
314#102—62no—
315#102—62no—
316#102—62no—
317C22⋊C8—32no—
318C22⋊C8—32no—
319C22⋊C8#6712yes—
320C4×C8—31no—
321C4×C8—31no—
322C4×C8—31no—
323C4×C8—31no—
324C4×C8C22×D311yescharacteristic
325C4×C8C22×D311yescharacteristic
326C23×C3—41no—
327C23×C3#12711yescharacteristic
328C22×D3—41no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×Q8—82no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—82no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—82no—
344C2×C3×C4—82no—
345C2×C3×C4#12711yescharacteristic
346C2×C3×C4#12711yescharacteristic
347C2×C3×C4#12711yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351#67—82no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355D12—42no—
356D12—42no—
357D12—42no—
358C4×D3—82no—
359Dic6—42no—
360Dic6—42no—
361Dic6—42no—
362C3×C8—22no—
363C3×C8—22no—
364C3⋊C8—22no—
365C3⋊C8—22no—
366C3⋊C8—22no—
367C3⋊C8—22no—
368#41—82no—
369#41—242no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—41no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—82no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—41no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—121no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—121no—
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4C2×#6711yescharacteristic
392Q16—242no—
393Q16—242no—
394Q16—242no—
395SD16—242no—
396SD16—242no—
397D8—242no—
398C2×C8—32no—
399C2×C8—32no—
400C2×C8—32no—
401C2×C8—32no—
402C2×C8C2×#6712yes—
403C2×C8C2×#6712yes—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—121no—
419C4⋊C4—121no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—121no—
423C22⋊C4—121no—
424C22⋊C4—121no—
425C42—121no—
426C42D3×D411yescharacteristic
427C42D3×D411yescharacteristic
428C42C2×#6711yescharacteristic
429C42C23×D311yescharacteristic
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3C2×#12711yescharacteristic
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—81no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—81no—
446C3×C4#57612yes—
447C3×C4#57712yes—
448C3×C4#57812yes—
449Dic3—81no—
450Dic3—81no—
451Dic3—81no—
452C23—41no—
453C23—121no—
454C23#94011yescharacteristic
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—82no—
459Q8—122no—
460Q8—122no—
461Q8—122no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—82no—
466D4—122no—
467D4—122no—
468D4—122no—
469D4—242no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—242no—
482C2×C4#94011yescharacteristic
483C2×C4#94011yescharacteristic
484C2×C4#94011yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×D3×D411yescharacteristic
487C2×C4C22×#6711yescharacteristic
488C8—22no—
489C8—22no—
490C8—62no—
491C8—62no—
492C8—62no—
493C8—62no—
494C2×C3—41no—
495C2×C3—161no—
496C2×C3#315411yescharacteristic
497C2×C3#315511yescharacteristic
498C2×C3#315611yescharacteristic
499D3—161no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22C2×#94011yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—241no—
517C4—241no—
518C4—241no—
519C4#575712yes—
520C4#577912yes—
521C4#583712yes—
522C3#1974111yescharacteristic, complemented, Sylow
523C2—41no—
524C2—161no—
525C2—481no—
526C2#8340211yescharacteristic
527C2#8383011yescharacteristic
528C2#8397511yescharacteristic
subgroupG/H
centerC22C2×#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3226C2
socleC22×C3C2×#127
prime radical#83303C2
hypercenter#3226D3
hypocenterC3#19741
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
1. C3×#3226C2semidirect
3. #83272C2semidirect
4. #83222C2semidirect
5. #83216C2semidirect
6. #79306C2semidirect
7. #71368C2semidirect
8. #71366C2semidirect
9. #71363C2semidirect
10. #71360C2semidirect
11. #71193C2semidirect
12. #71192C2semidirect
13. #71186C2semidirect
14. #71182C2semidirect
15. #70744C2semidirect
66. #3226D3semidirect
522. C3#19741semidirect

Maximal subgroups

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

subgroupindex
C3×#32262
#833032
#832722
#832222
#832162
#793062
#713682
#713662
#713632
#713602
#711932
#711922
#711862
#711822
#707442
#197413