Finite Group Databasebeta

← #10848871

Subgroups — #10848871

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
1C3×#3511C211yesmaximal, characteristic
2#83288C211yesmaximal, characteristic, complemented
3#83252C211yesmaximal, characteristic, complemented
4#82533C211yesmaximal, characteristic, complemented
5#71881C214yesmaximal, complemented
6#71815C211yesmaximal, characteristic, complemented
7#71790C211yesmaximal, characteristic, complemented
8#71666C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic, complemented
10#25615—31nomaximal, Sylow
11C3×#556—21no—
12C3×#555—21no—
13C3×#554—21no—
14C3×#553—22no—
15C3×#548—22no—
16C3×#547—21no—
17C3×#542—22no—
18C3×#537C2212yes—
19C3×#533—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C3×#520C2212yes—
24C3×C4×#41C2211yescharacteristic
25C2×#951C2214yes—
26C2×#951C2212yes—
27C2×#869C2211yescharacteristic
28C2×#837C2212yes—
29C2×#837C2214yes—
30#5229—22no—
31#5092—21no—
32#5076—21no—
33#5060—22no—
34#5031—21no—
35#5031—21no—
36#5019—21no—
37#5019—21no—
38#5004C2212yes—
39#4977—22no—
40#4973C2214yes—
41C42×Dic3C2211yescharacteristic
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4532C2212yes—
45#4530C2214yes—
46#3511D311yescharacteristic
47#3152—31no—
48#3039—31no—
49#2862—31no—
50#2007—34no—
51#1997—31no—
52#1962—31no—
53#1882—34no—
54C4×#104—31no—
55C2×C3×#41—22no—
56C2×C3×#41C2311yescharacteristic
57C2×C3×#41C2312yes—
58C22×C3×Q8D412yes—
59C22×Dic6C2312yes—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8D411yescharacteristic
65C3×#127D411yescharacteristic
66C3×#126—24no—
67C3×#125—22no—
68C3×#124—22no—
69C3×#124—24no—
70C3×#124D412yes—
71C3×#123—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—21no—
80C3×#121—21no—
81C3×#121—24no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D411yescharacteristic
85C3×C4×Q8D411yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—24no—
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4D411yescharacteristic
94C3×C4×D4D411yescharacteristic
95C3×#117—22no—
96C3×#117—22no—
97C3×#117—21no—
98C3×#117C2311yescharacteristic
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4D412yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104#951—28no—
105#951—44no—
106#951—44no—
107C2×#216—22no—
108C2×#216—24no—
109C2×#214—22no—
110C2×#214—24no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2312yes—
116C2×#208C2312yes—
117C2×#208C2312yes—
118#869—21no—
119#869—21no—
120#869—21no—
121#869—21no—
122#837—24no—
123#837—44no—
124C3⋊#95—28no—
125#810—22no—
126#810—22no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)—21no—
129#803—21no—
130C4×C3⋊C8—21no—
131C2×#137—34no—
132C2×#137—32no—
133#556—21no—
134#555—21no—
135#554—21no—
136#553—22no—
137#548—22no—
138#547—21no—
139#542—22no—
140#537C2×D312yes—
141#533—21no—
142#532—21no—
143#532C2×D311yescharacteristic
144C2×C4⋊Q8—31no—
145C2×C4⋊Q8C2×D311yescharacteristic
146#520C2×D312yes—
147C4×#41C2×D311yescharacteristic
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—34no—
161#391—62no—
162#382—62no—
163C43—31no—
164#343—34no—
165#339—32no—
166C3×#41—22no—
167C3×#41—24no—
168C3×#41—24no—
169C3×#41—22no—
170C3×#41—84no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—44no—
174C2×C3×Q8C2×D412yes—
175C2×C3×Q8C2×D411yescharacteristic
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2×D411yescharacteristic
183C22×C3×C4—24no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—22no—
193C22×Dic3—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—24no—
197C2×Dic6—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4C2×D411yescharacteristic
209C3×C4⋊C4C2×D411yescharacteristic
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—21no—
214C3×C22⋊C4—21no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C42—24no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42C2×D411yescharacteristic
223C3×C42C2×D411yescharacteristic
224C3×C42C2×D411yescharacteristic
225C3×C42C2×D411yescharacteristic
226#216—48no—
227#216—44no—
228#214—48no—
229#214—44no—
230C4⋊Dic3—41no—
231C4⋊Dic3—41no—
232C4⋊Dic3—41no—
233C4⋊Dic3—41no—
234C4×Dic3—24no—
235C4×Dic3—22no—
236C4×Dic3—22no—
237C4×Dic3—42no—
238#208—22no—
239#208—24no—
240#208#4114yescomplemented
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×#41—22no—
244C2×#41C22×D311yescharacteristic
245C2×#41C22×D312yes—
246C22×Q8—32no—
247C22×Q8#6712yes—
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—61no—
262C4⋊Q8—61no—
263C4⋊Q8—61no—
264C4⋊Q8—61no—
265C4⋊Q8#6711yescharacteristic
266#127#6711yescharacteristic
267#126—24no—
268#125—22no—
269#124—22no—
270#124—24no—
271#124#6712yes—
272#123—24no—
273C22⋊Q8—22no—
274C22⋊Q8—24no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—21no—
281#121—21no—
282#121—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8#6711yescharacteristic
286C4×Q8#6711yescharacteristic
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—24no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4#6711yescharacteristic
295C4×D4#6711yescharacteristic
296#117—22no—
297#117—22no—
298#117—21no—
299#117C22×D311yescharacteristic
300C2×C4⋊C4—22no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4#6712yes—
304C2×C4⋊C4C22×D312yes—
305C2×C42—21no—
306C2×C42—32no—
307C2×C42—61no—
308C2×C42—61no—
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—68no—
319C23×C3—21no—
320C23×C3—22no—
321C23×C3C22×D411yescharacteristic
322C3×Q8—21no—
323C3×Q8—21no—
324C3×Q8—24no—
325C3×Q8—44no—
326C3×Q8—82no—
327C3×D4—21no—
328C3×D4—21no—
329C3×D4—24no—
330C3×D4—42no—
331C3×D4—44no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—84no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—24no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—44no—
348C2×C3×C4—84no—
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4#12011yescharacteristic
351C2×C3×C4#12011yescharacteristic
352C2×C3×C4#12011yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4112yes—
356C2×C3×C4H(5,𝔽2)12yes—
357C2×Dic3—24no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—44no—
362Dic6—42no—
363Dic6—42no—
364Dic6—44no—
365C3⋊C8—24no—
366C3⋊C8—42no—
367#41—22no—
368#41—24no—
369#41—24no—
370#41—22no—
371#41—84no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—44no—
375C2×Q8—64no—
376C2×Q8—64no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8C2×#6712yes—
380C2×Q8C2×#6711yescharacteristic
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4C2×#6712yes—
387C2×D4C2×#6711yescharacteristic
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—62no—
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—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—121no—
419C4⋊C4—121no—
420C4⋊C4—121no—
421C4⋊C4—121no—
422C4⋊C4C2×#6711yescharacteristic
423C4⋊C4C2×#6711yescharacteristic
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C42—24no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—64no—
437C42—62no—
438C42—62no—
439C42—122no—
440C42C2×#6711yescharacteristic
441C42C2×#6711yescharacteristic
442C42C2×#6711yescharacteristic
443C42C2×#6711yescharacteristic
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—42no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—42no—
450C22×C3—41no—
451C22×C3—42no—
452C22×C3C2×#12011yescharacteristic
453C22×C3#55012yes—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—42no—
457C3×C4—41no—
458C3×C4—41no—
459C3×C4—44no—
460C3×C4—41no—
461C3×C4#45512yes—
462C3×C4#45612yes—
463C3×C4#53211yescharacteristic
464C3×C4#53511yescharacteristic
465C3×C4#54612yes—
466Dic3—44no—
467Dic3—82no—
468C23—21no—
469C23—22no—
470C23C22×#6711yescharacteristic
471Q8—24no—
472Q8—21no—
473Q8—21no—
474Q8—44no—
475Q8—82no—
476Q8—124no—
477Q8—122no—
478Q8—122no—
479D4—24no—
480D4—21no—
481D4—21no—
482D4—42no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—84no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—64no—
501C2×C4—84no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—124no—
505C2×C4—122no—
506C2×C4#95311yescharacteristic
507C2×C4#95311yescharacteristic
508C2×C4#95311yescharacteristic
509C2×C4#95311yescharacteristic
510C2×C4D3×#4112yes—
511C2×C4#100912yes—
512C2×C4C22×#6711yescharacteristic
513C2×C4C22×#6711yescharacteristic
514C8—64no—
515C8—122no—
516C2×C3—22no—
517C2×C3—81no—
518C2×C3—82no—
519C2×C3#243111yescharacteristic
520C2×C3C2×#45511yescharacteristic
521C2×C3C2×#45611yescharacteristic
522C22—21no—
523C22—21no—
524C22—41no—
525C22—42no—
526C22—41no—
527C22—42no—
528C22—41no—
529C22—42no—
530C22#583412yes—
531C22C2×#95311yescharacteristic
532C4—21no—
533C4—21no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—44no—
538C4—41no—
539C4—124no—
540C4—242no—
541C4#524212yes—
542C4#524612yes—
543C4#582912yes—
544C4#583011yescharacteristic
545C4#583211yescharacteristic
546C3#2561511yescharacteristic, complemented, Sylow
547C2—22no—
548C2—81no—
549C2—82no—
550C2#8409211yescharacteristic
551C2C2×#524211yescharacteristic
552C2C2×#524611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3511C2
socleC22×C3C2×#120
hypercenter#3511D3
hypocenterC3#25615
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
2. #83288C2semidirect
3. #83252C2semidirect
4. #82533C2semidirect
5. #71881C2semidirect
6. #71815C2semidirect
7. #71790C2semidirect
8. #71666C2semidirect
9. C4×#837C2semidirect
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
C3×#35112
#832882
#832522
#825332
#718812
#718152
#717902
#716662
C4×#8372
#256153