Finite Group Databasebeta

← #10786000

Subgroups — #10786000

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3302C211yesmaximal, characteristic
2#83680C211yesmaximal, characteristic, complemented
3#83517C211yesmaximal, characteristic
4#79632C211yesmaximal, characteristic, complemented
5#71932C211yesmaximal, characteristic, complemented
6#71913C211yesmaximal, characteristic
7#71240C211yesmaximal, characteristic, complemented
8#71217C211yesmaximal, characteristic, complemented
9#71113C211yesmaximal, characteristic, complemented
10#71101C211yesmaximal, characteristic, complemented
11#71067C211yesmaximal, characteristic, complemented
12#71056C211yesmaximal, characteristic, complemented
13#70895C211yesmaximal, characteristic, complemented
14#70851C211yesmaximal, characteristic, complemented
15#70839C211yesmaximal, characteristic, complemented
16#19853—31nomaximal, Sylow
17C3×#556C2211yescharacteristic
18C3×#547C2211yescharacteristic
19#5570C2211yescharacteristic
20#5543C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#485—21no—
23C3×#474C2211yescharacteristic
24C3×#467—21no—
25C3×#462C2211yescharacteristic
26C3×#452—21no—
27C3×C4×SD16C2211yescharacteristic
28C3×C8×D4C2211yescharacteristic
29C3×#416—21no—
30Dic3×SD16—21no—
31#5094C2211yescharacteristic
32#5093C2211yescharacteristic
33#5092C2211yescharacteristic
34#5090C2211yescharacteristic
35#5080C2211yescharacteristic
36#5080C2211yescharacteristic
37#5060—21no—
38#5054—21no—
39#5041—21no—
40#5034—21no—
41#5031—21no—
42#5030—21no—
43#5024C2211yescharacteristic
44#5021C2211yescharacteristic
45#5019C2211yescharacteristic
46C4×#214C2211yescharacteristic
47D4×C3⋊C8C2211yescharacteristic
48#5008C2211yescharacteristic
49#5007C2211yescharacteristic
50#4971—21no—
51C2×#808—21no—
52#4858—21no—
53#4799—21no—
54#4763—21no—
55#4680C2211yescharacteristic
56C8×Dic6C2211yescharacteristic
57#4547C2211yescharacteristic
58#4547C2211yescharacteristic
59#4535C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4486C2211yescharacteristic
63#4484C2211yescharacteristic
64#4481C2211yescharacteristic
65#4479C2211yescharacteristic
66#4457C2211yescharacteristic
67#4454C2211yescharacteristic
68#3398—31no—
69#3381—31no—
70#3335—31no—
71#3302D311yescharacteristic
72#1716—31no—
73#1692—31no—
74#1680—31no—
75#1671—31no—
76#1669—31no—
77#1654—31no—
78#1653—31no—
79#1644—31no—
80#1638—31no—
81#1582—31no—
82SD16⋊C8—31no—
83C2×C3×#41—21no—
84C2×C3×Q16—21no—
85C2×C3×SD16D411yescharacteristic
86C22×C3×C8—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×#126—41no—
90C3×#124—21no—
91C3×#123—41no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—21no—
94C3×#121—21no—
95C3×C4×Q8—21no—
96C3×C4×Q8C2311yescharacteristic
97C3×C4×D4C2311yescharacteristic
98C3×C4×D4C2311yescharacteristic
99C3×#117—21no—
100C2×C3×C4⋊C4—21no—
101Q8×Dic3—21no—
102C2×#216—21no—
103C2×#216—21no—
104C2×#216—21no—
105D4×Dic3—21no—
106C2×#214—21no—
107C2×#214D411yescharacteristic
108C2×C4⋊Dic3—21no—
109#924—21no—
110C22×C3⋊C8—21no—
111#890—21no—
112#890—21no—
113#890—41no—
114#878—41no—
115#869—21no—
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6—21no—
119C4×Dic6C2311yescharacteristic
120C3×#107—21no—
121C3×#106D411yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×#103—21no—
124C3×#103—21no—
125C3×#103—21no—
126C3×#103D411yescharacteristic
127C3×#102—21no—
128C3×#102D411yescharacteristic
129C3×C22⋊C8—21no—
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#835—21no—
133#832—21no—
134#832—21no—
135#832D411yescharacteristic
136#830—21no—
137#817—21no—
138#817—21no—
139Dic3⋊C8—21no—
140C8×Dic3—21no—
141#810—21no—
142#810—21no—
143#810—41no—
144#810D411yescharacteristic
145#808—21no—
146#808—21no—
147#808—21no—
148#808—21no—
149#808—41no—
150#808D411yescharacteristic
151#807—21no—
152#807—21no—
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157#803—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#557—31no—
161#556C2×D311yescharacteristic
162#554—31no—
163#547C2×D311yescharacteristic
164#499—61no—
165#498—31no—
166#498—31no—
167#498—31no—
168#497—61no—
169#494—31no—
170#487—31no—
171#487C2×D311yescharacteristic
172#485—21no—
173#483—61no—
174#481—61no—
175#479—61no—
176#478—61no—
177#477—31no—
178#476—31no—
179#475—31no—
180#474—31no—
181#474C2×D311yescharacteristic
182#467—21no—
183#465—61no—
184#464—61no—
185#462—31no—
186#462—61no—
187#462C2×D311yescharacteristic
188#461—31no—
189#452—21no—
190#446—31no—
191C8×Q8—31no—
192#441—61no—
193C4×SD16—31no—
194C4×SD16—61no—
195C4×SD16C2×D311yescharacteristic
196C8×D4—31no—
197C8×D4C2×D311yescharacteristic
198C2×#106—61no—
199#416—21no—
200#335—31no—
201#334—31no—
202#333—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8—31no—
207Q8⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C3×#41—42no—
212C3×#41—81no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—41no—
217C2×C3×D4C2×D411yescharacteristic
218C2×C3×D4C2×D411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—41no—
222C22×Dic3—41no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D411yescharacteristic
226C3×Q16—81no—
227C3×SD16—22no—
228C2×C3×C8—21no—
229C2×C3×C8—81no—
230C2×C3×C8C2×D411yescharacteristic
231C2×C3×C8C2×D411yescharacteristic
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C4⋊C4C2×D411yescharacteristic
241C3×C4⋊C4C2×D411yescharacteristic
242C3×C22⋊C4—21no—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—41no—
245C3×C22⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C42—21no—
248C3×C42—41no—
249C3×C42C2411yescharacteristic
250#217—41no—
251#216—42no—
252#216—81no—
253#216—81no—
254#214—22no—
255#214—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—41no—
261C4⋊Dic3—41no—
262C4⋊Dic3—41no—
263C4⋊Dic3C2×D411yescharacteristic
264Dic3⋊C4—21no—
265Dic3⋊C4—41no—
266Dic3⋊C4—41no—
267C4×Dic3—21no—
268C4×Dic3—41no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—42no—
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×#41—21no—
277C2×Q16—21no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×Q16—61no—
281C2×SD16—31no—
282C2×SD16—61no—
283C2×SD16#6711yescharacteristic
284C22×C8—21no—
285C22×C8—61no—
286C4⋊Q8—31no—
287C4⋊Q8—31no—
288C4⋊Q8—61no—
289C4⋊Q8C22×D311yescharacteristic
290C4⋊Q8C22×D311yescharacteristic
291#126—41no—
292#125—61no—
293#125—61no—
294#125—121no—
295#124—21no—
296#123—41no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—61no—
300C22⋊Q8—121no—
301#121—21no—
302C4×Q8—21no—
303C4×Q8—31no—
304C4×Q8—61no—
305C4×Q8—61no—
306C4×Q8C22×D311yescharacteristic
307C4×D4—61no—
308C4×D4C22×D311yescharacteristic
309C4×D4C22×D311yescharacteristic
310#117—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—61no—
313#107—21no—
314#107—61no—
315#107—61no—
316#106—31no—
317#106—61no—
318#106—61no—
319#106—61no—
320#106—61no—
321#106—61no—
322#106—61no—
323#106—121no—
324#106#6711yescharacteristic
325C4⋊C8—31no—
326C4⋊C8—31no—
327C4⋊C8—31no—
328C4⋊C8—61no—
329C4⋊C8—61no—
330C4⋊C8C22×D311yescharacteristic
331#103—21no—
332#103—21no—
333#103—21no—
334#103—31no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—121no—
340#103#6711yescharacteristic
341#102—21no—
342#102—31no—
343#102—61no—
344#102—61no—
345#102#6711yescharacteristic
346C22⋊C8—21no—
347C22⋊C8—61no—
348#97—61no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C23×C3—21no—
354C23×C3—21no—
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—82no—
365C3×D4#13712yes—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—82no—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—161no—
385Dic6—21no—
386Dic6—21no—
387Dic6—41no—
388Dic6—41no—
389Dic6—81no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—41no—
393C3⋊C8—21no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C3⋊C8#13512yes—
399#41—42no—
400#41—81no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—31no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8C2×#6711yescharacteristic
407C2×D4—41no—
408C2×D4D3×D411yescharacteristic
409C2×D4C2×#6711yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—121no—
414Q16—81no—
415Q16—122no—
416Q16—241no—
417Q16—241no—
418SD16—22no—
419SD16—62no—
420SD16—241no—
421C2×C8—21no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—81no—
429C2×C8—122no—
430C2×C8C2×#6711yescharacteristic
431C2×C8C2×#6711yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—31no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C4⋊C4—121no—
448C4⋊C4—121no—
449C4⋊C4—121no—
450C4⋊C4D3×D411yescharacteristic
451C4⋊C4C2×#6711yescharacteristic
452C4⋊C4C2×#6711yescharacteristic
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—121no—
459C42—21no—
460C42—41no—
461C42—61no—
462C42—121no—
463C42C23×D311yescharacteristic
464C22×C3—22no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3D4211yescharacteristic
471C3×C4—21no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4C2×#13511yescharacteristic
479C3×C4C2×#13711yescharacteristic
480C3×C4#57711yescharacteristic
481C3×C4#57711yescharacteristic
482Dic3—41no—
483Dic3—41no—
484Dic3—81no—
485Dic3—81no—
486C23—21no—
487C23—21no—
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—61no—
495Q8—61no—
496Q8—121no—
497Q8—121no—
498Q8—241no—
499D4—21no—
500D4—21no—
501D4—21no—
502D4—82no—
503D4#95112yes—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—82no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—481no—
520C2×C4C2×D3×D411yescharacteristic
521C2×C4#100411yescharacteristic
522C2×C4C22×#6711yescharacteristic
523C8—21no—
524C8—21no—
525C8—32no—
526C8—41no—
527C8—61no—
528C8—121no—
529C8—121no—
530C8—121no—
531C8—121no—
532C2×C3—42no—
533C2×C3—81no—
534C2×C3#326611yescharacteristic
535C2×C3#328411yescharacteristic
536C2×C3#328911yescharacteristic
537C22—22no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22D4×#6711yescharacteristic
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—121no—
552C4—121no—
553C4—241no—
554C4—241no—
555C4D3×#13511yescharacteristic
556C4#577011yescharacteristic
557C4C2×#95111yescharacteristic
558C4#583611yescharacteristic
559C3#1985311yescharacteristic, complemented, Sylow
560C2—42no—
561C2—81no—
562C2#8350611yescharacteristic
563C2#8392711yescharacteristic
564C2#8401911yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3302C2
socleC22×C3D42
prime radicalC3×#547C22
hypercenter#3302D3
hypocenterC3#19853
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #3302

Complements

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

NH ≅ G/Ntype
2. #83680C2semidirect
4. #79632C2semidirect
5. #71932C2semidirect
7. #71240C2semidirect
8. #71217C2semidirect
9. #71113C2semidirect
10. #71101C2semidirect
11. #71067C2semidirect
12. #71056C2semidirect
13. #70895C2semidirect
14. #70851C2semidirect
15. #70839C2semidirect
559. C3#19853semidirect

Maximal subgroups

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

subgroupindex
C3×#33022
#836802
#835172
#796322
#719322
#719132
#712402
#712172
#711132
#711012
#710672
#710562
#708952
#708512
#708392
#198533