Finite Group Databasebeta

← #10848878

Subgroups — #10848878

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86205C211yesmaximal, characteristic
2C3×#2862C211yesmaximal, characteristic, complemented
3#83209C211yesmaximal, characteristic, complemented
4#83203C211yesmaximal, characteristic, complemented
5#71790C211yesmaximal, characteristic, complemented
6#69694C214yesmaximal, complemented
7#69693C214yesmaximal, complemented
8#69395C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic, complemented
10#25615—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5817—22no—
14C2×#942C2214yes—
15C2×#942C2212yes—
16#5734—21no—
17#5733—22no—
18#5732—21no—
19#5730—21no—
20#5728—21no—
21#5726—21no—
22#5725—21no—
23#5514—22no—
24#5510C2212yes—
25#5504C2212yes—
26#5477C2211yescharacteristic
27C4×#235C2211yescharacteristic
28C3×#391—22no—
29C3×#391C2214yes—
30C3×#382—22no—
31C3×C43C2211yescharacteristic
32#5225C2212yes—
33#5092—21no—
34#5090—21no—
35#5090—21no—
36#5086—22no—
37#5085—21no—
38#5083—21no—
39#5083—21no—
40C2×#805C2212yes—
41C2×#805C2214yes—
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4530C2214yes—
45#4523C2212yes—
46#3511—31no—
47#3152—31no—
48#3039—31no—
49#2862D311yescharacteristic, complemented
50#2007—34no—
51#1997—31no—
52#1962—31no—
53#1882—34no—
54C4×#104—31no—
55C22×C3×Q8D412yes—
56C22×C3×Q8C2312yes—
57C2×#239—22no—
58C2×#235C2311yescharacteristic
59C2×#235C2312yes—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76#947—22no—
77#947—24no—
78#946—22no—
79#946—24no—
80Q8×Dic3—22no—
81C2×#216—24no—
82C2×#216—22no—
83#942—28no—
84#942—44no—
85#942—44no—
86C2×#215—22no—
87C2×#215—24no—
88#930—21no—
89#930—21no—
90#930—24no—
91C4×#67—21no—
92C4×#67—21no—
93C4×#67—24no—
94#926C2311yescharacteristic
95C2×#208C2312yes—
96C2×#208C2312yes—
97#898—24no—
98#897—22no—
99#897—22no—
100#895—22no—
101#895—22no—
102#895—22no—
103#894—24no—
104#893—22no—
105#893—22no—
106#892—22no—
107#892—22no—
108D3×C4⋊C4—22no—
109#890—22no—
110#887—22no—
111#875D412yes—
112#874D411yescharacteristic
113C4×D12—21no—
114C4×D12—21no—
115C4×D12D411yescharacteristic
116C4×D12D411yescharacteristic
117#872—21no—
118C42×D3—21no—
119#869D411yescharacteristic
120C4×Dic6D411yescharacteristic
121C4×Dic6D411yescharacteristic
122C3×#95—28no—
123#835—22no—
124#835—22no—
125#805—24no—
126#805—44no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)—21no—
129#803—21no—
130C4×C3⋊C8—21no—
131C2×#137—34no—
132C2×#137—32no—
133#556—61no—
134#555—61no—
135#554—61no—
136#553—62no—
137#548—62no—
138#547—61no—
139#542—62no—
140#537—32no—
141#533—61no—
142#532—31no—
143#532—61no—
144C2×C4⋊Q8C2×D311yescharacteristic
145C2×C4⋊Q8C2×D311yescharacteristic
146#520—32no—
147C4×#41—31no—
148#497—62no—
149#494—61no—
150#492—61no—
151#462—61no—
152#462—61no—
153#461—61no—
154#461—61no—
155#423—31no—
156C2×#104—32no—
157C2×#104—34no—
158#419—32no—
159C4×M16—31no—
160#391—22no—
161#391C2×D314yes—
162#382—22no—
163C43C2×D311yescharacteristic
164#343—34no—
165#339—32no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—24no—
171C2×C3×Q8—24no—
172C2×C3×Q8—44no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—21no—
180C2×#67—22no—
181#239—84no—
182#235—22no—
183#235—24no—
184#235—24no—
185#235—22no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—24no—
189C2×D12C2×D411yescharacteristic
190C2×D12C2×D412yes—
191C2×C4×D3—21no—
192C2×C4×D3—21no—
193C2×C4×D3—21no—
194C2×C4×D3—22no—
195C2×C4×D3—24no—
196C2×Dic6C2×D411yescharacteristic
197C2×Dic6C2×D412yes—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4—41no—
206C3×C42—24no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—42no—
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214#217—21no—
215#216—48no—
216#216—44no—
217#215—48no—
218#215—44no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—21no—
223(C2×D3)⋊C4—21no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4—22no—
226C4⋊Dic3—24no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3C2×D411yescharacteristic
230C4⋊Dic3C2×D411yescharacteristic
231Dic3⋊C4—24no—
232Dic3⋊C4—21no—
233Dic3⋊C4—21no—
234C4×Dic3—24no—
235C4×Dic3—21no—
236C4×Dic3—21no—
237C4×Dic3—21no—
238#208—22no—
239#208—24no—
240#208#4114yescomplemented
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×#41—31no—
244C2×#41—32no—
245C2×#41—62no—
246C22×Q8#6712yes—
247C22×Q8C22×D312yes—
248#137—68no—
249#137—124no—
250#137—124no—
251C2×Q16—62no—
252C2×Q16—64no—
253C2×SD16—62no—
254C2×SD16—64no—
255C2×M16—32no—
256C2×M16—32no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—31no—
266#127—31no—
267#126—64no—
268#125—62no—
269#124—32no—
270#124—62no—
271#124—64no—
272#123—64no—
273C22⋊Q8—62no—
274C22⋊Q8—62no—
275C22⋊Q8—62no—
276C22⋊Q8—64no—
277#121—62no—
278#121—62no—
279#121—62no—
280#121—61no—
281#121—61no—
282#121—64no—
283C4×Q8—31no—
284C4×Q8—31no—
285C4×Q8—62no—
286C4×Q8—62no—
287C4×D4—31no—
288C4×D4—31no—
289C4×D4—62no—
290C4×D4—62no—
291C4×D4—61no—
292C4×D4—61no—
293C4×D4—64no—
294C4×D4—61no—
295C4×D4—61no—
296#117—31no—
297#117—62no—
298#117—62no—
299#117—61no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4#6712yes—
304C2×C4⋊C4C22×D312yes—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—61no—
308C2×C42C22×D312yes—
309C2×C42C22×D311yescharacteristic
310C4⋊C8—61no—
311C4⋊C8—61no—
312#104—64no—
313#104—124no—
314#103—62no—
315#103—62no—
316#97—61no—
317C4×C8—61no—
318#95—28no—
319C23×C3C22×D411yescharacteristic
320C22×D3—21no—
321C22×D3—22no—
322C3×Q8—44no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—44no—
326C3×Q8—82no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4#12011yescharacteristic
337C2×C3×C4#12011yescharacteristic
338C2×C3×C4#12011yescharacteristic
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4112yes—
343C2×C3×C4H(5,𝔽2)12yes—
344#67—42no—
345#67—44no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—22no—
349C2×Dic3—21no—
350D12—21no—
351D12—21no—
352D12—24no—
353D12—42no—
354D12—42no—
355D12—84no—
356C4×D3—42no—
357C4×D3—42no—
358C4×D3—42no—
359C4×D3—42no—
360C4×D3—42no—
361C4×D3—84no—
362Dic6—21no—
363Dic6—21no—
364Dic6—24no—
365C3⋊C8—24no—
366C3⋊C8—42no—
367#41—62no—
368#41—64no—
369#41—64no—
370#41—62no—
371#41—244no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—31no—
379C2×Q8—32no—
380C2×Q8—44no—
381C2×D4—31no—
382C2×D4—32no—
383C2×D4—64no—
384C2×D4—61no—
385C2×D4—62no—
386C2×D4—61no—
387C2×D4—61no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—64no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—62no—
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6714yes—
398C22×C4C23×D311yescharacteristic
399Q16—128no—
400Q16—124no—
401SD16—128no—
402SD16—124no—
403M16—34no—
404M16—62no—
405M16—64no—
406C2×C8—62no—
407C2×C8—62no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—31no—
413C4⋊C4—31no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—64no—
419C4⋊C4—64no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C22⋊C4—64no—
425C22⋊C4—64no—
426C22⋊C4—64no—
427C22⋊C4—61no—
428C22⋊C4—61no—
429C22⋊C4—61no—
430C22⋊C4—62no—
431C22⋊C4—62no—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—42no—
436C42—64no—
437C42—61no—
438C42—61no—
439C42—61no—
440C42D3×D411yescharacteristic
441C42D3×D411yescharacteristic
442C42D3×D411yescharacteristic
443C42D3×D411yescharacteristic
444C22×C3—21no—
445C22×C3—21no—
446C22×C3C2×#12011yescharacteristic
447C22×C3#55012yes—
448C2×D3—42no—
449C2×D3—41no—
450C2×D3—41no—
451C2×D3—42no—
452C2×D3—41no—
453C2×D3—42no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—44no—
457C3×C4—44no—
458C3×C4—82no—
459C3×C4#45512yes—
460C3×C4#45612yes—
461C3×C4#53211yescharacteristic
462C3×C4#53511yescharacteristic
463C3×C4#54612yes—
464Dic3—41no—
465Dic3—42no—
466Dic3—41no—
467Dic3—41no—
468C23—61no—
469C23—62no—
470C23C22×#6711yescharacteristic
471Q8—44no—
472Q8—42no—
473Q8—42no—
474Q8—44no—
475Q8—61no—
476Q8—61no—
477Q8—64no—
478Q8—82no—
479D4—61no—
480D4—61no—
481D4—64no—
482D4—122no—
483D4—124no—
484D4—122no—
485D4—122no—
486D4—244no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—244no—
506C2×C4#93711yescharacteristic
507C2×C4#93711yescharacteristic
508C2×C4#93711yescharacteristic
509C2×C4#93711yescharacteristic
510C2×C4C2×D3×D411yescharacteristic
511C2×C4C2×D3×D411yescharacteristic
512C2×C4D3×#4112yes—
513C2×C4#100912yes—
514C8—64no—
515C8—122no—
516C2×C3—22no—
517C2×C3#243111yescharacteristic
518C2×C3C2×#45511yescharacteristic
519C2×C3C2×#45611yescharacteristic
520D3—81no—
521D3—82no—
522C22—21no—
523C22—21no—
524C22—121no—
525C22—122no—
526C22—122no—
527C22—121no—
528C22—121no—
529C22—122no—
530C22C2×#93711yescharacteristic
531C22#583412yes—
532C4—21no—
533C4—21no—
534C4—44no—
535C4—44no—
536C4—82no—
537C4—121no—
538C4—122no—
539C4—121no—
540C4—121no—
541C4#506112yes—
542C4#509512yes—
543C4#566811yescharacteristic
544C4#567411yescharacteristic
545C4#567612yes—
546C3#2561511yescharacteristic, complemented, Sylow
547C2—22no—
548C2—241no—
549C2—242no—
550C2C2×#506111yescharacteristic
551C2#8295811yescharacteristic
552C2C2×#509511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2862C2
socleC22×C3C2×#120
prime radical#86205C2
hypercenter#2862D3
hypocenterC3#25615
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
2. C3×#2862C2semidirect
3. #83209C2semidirect
4. #83203C2semidirect
5. #71790C2semidirect
6. #69694C2semidirect
7. #69693C2semidirect
8. #69395C2semidirect
9. C4×#805C2semidirect
49. #2862D3semidirect
240. #208#41semidirect
546. C3#25615semidirect

Maximal subgroups

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

subgroupindex
#862052
C3×#28622
#832092
#832032
#717902
#696942
#696932
#693952
C4×#8052
#256153