Finite Group Databasebeta

← #10686702

Subgroups — #10686702

Order 768.

906 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85718C211yesmaximal, characteristic
2C3×#1436C211yesmaximal, characteristic, complemented
3#71552C211yesmaximal, characteristic, complemented
4#71532C212yesmaximal, complemented
5#69398C212yesmaximal, complemented
6#8347—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C22×C3×C42C2211yescharacteristic
9#5791—21no—
10#5791—21no—
11#5791C2212yes—
12#5515—21no—
13#5515C412yescomplemented
14#5514—22no—
15#5513—22no—
16#5511—22no—
17#5510C411yescharacteristic, complemented
18#5507—22no—
19#5504—21no—
20#5500C411yescharacteristic, complemented
21C2×C3×#95—21no—
22C2×C3×#95—21no—
23C2×C3×#95C2211yescharacteristic
24#5207C2212yes—
25C2×#805—22no—
26#4528—21no—
27#4528—21no—
28#4523—22no—
29#4452—22no—
30#4451—22no—
31#3492—31no—
32#1896—32no—
33#1504—32no—
34#1463—31no—
35#1436D311yescharacteristic, complemented
36C23×C3×C4—21no—
37C23×C3×C4—21no—
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C2×#235—21no—
42C2×#235—21no—
43C2×#235D412yes—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—22no—
51C2×C3×C42—42no—
52#953—22no—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57#930C2×C412yes—
58#930D412yes—
59#929—22no—
60#929—22no—
61#929—22no—
62C4×#67—22no—
63C4×#67—22no—
64C4×#67—22no—
65C4×#67—22no—
66C4×#67—22no—
67C4×#67—22no—
68#926—21no—
69#926—21no—
70#926D412yes—
71#924—21no—
72#924—21no—
73#924—21no—
74#924—21no—
75#924C2×C412yes—
76#924D412yes—
77C2×#208—22no—
78#898—22no—
79#898—22no—
80#897—22no—
81#896—22no—
82#896—22no—
83#895—22no—
84#894—22no—
85#894—22no—
86#893—22no—
87#892—22no—
88D3×C4⋊C4—22no—
89#890—22no—
90#889—22no—
91#889—22no—
92#888—22no—
93#887—22no—
94C3×#95—24no—
95C3×#95—22no—
96C3×#95—42no—
97#830—22no—
98#830—22no—
99#805—44no—
100#554—62no—
101#553—62no—
102#552—62no—
103#547—62no—
104#544—31no—
105#539—32no—
106#539—61no—
107#537—31no—
108#525—32no—
109#525—61no—
110#525—61no—
111#520—61no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C42C2×D311yescharacteristic
114C2×#104—62no—
115#407—32no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95C2×D311yescharacteristic
119#339—62no—
120#328—62no—
121#327—62no—
122#324—61no—
123#324—61no—
124C24×C3C2×D411yescharacteristic
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4C22⋊C411yescharacteristic
150C22×C3×C4C22⋊C411yescharacteristic
151C22×C3×C4C22⋊C411yescharacteristic
152C22×C3×C4C22⋊C411yescharacteristic
153C22×C3×C4C22×C411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C2×#67—22no—
158C2×#67—22no—
159C2×#67—22no—
160#235—42no—
161#235—42no—
162#235—42no—
163#235—42no—
164#235—81no—
165#235—81no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12C22⋊C412yes—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6C22⋊C412yes—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C42—24no—
180C3×C42—42no—
181#217—22no—
182#217—22no—
183#217—22no—
184(C2×D3)⋊C4—22no—
185(C2×D3)⋊C4—22no—
186(C2×D3)⋊C4—22no—
187(C2×D3)⋊C4—22no—
188(C2×D3)⋊C4—22no—
189(C2×D3)⋊C4—22no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3C22⋊C412yes—
195C4⋊Dic3C22⋊C412yes—
196Dic3⋊C4—22no—
197Dic3⋊C4—22no—
198Dic3⋊C4—22no—
199Dic3⋊C4—22no—
200Dic3⋊C4—22no—
201Dic3⋊C4—22no—
202C4×Dic3—22no—
203C4×Dic3—22no—
204C4×Dic3—22no—
205#208—42no—
206#208—42no—
207C2×C3⋊C8—42no—
208C2×#41—32no—
209C2×#41—61no—
210C2×#41—61no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4#6711yescharacteristic
214C23×C4#6711yescharacteristic
215C23×C4C22×D311yescharacteristic
216C2×M16—62no—
217C4⋊Q8—62no—
218#126—62no—
219#126—62no—
220#125—62no—
221#125—62no—
222#125—62no—
223#123—62no—
224#123—62no—
225#123—62no—
226#123—62no—
227#123—62no—
228C22⋊Q8—32no—
229C22⋊Q8—32no—
230C22⋊Q8—62no—
231C22⋊Q8—62no—
232C22⋊Q8—62no—
233C22⋊Q8—61no—
234C22⋊Q8—61no—
235C22⋊Q8—61no—
236C22⋊Q8—61no—
237#121—32no—
238#121—32no—
239#121—62no—
240#121—61no—
241#121—61no—
242#121—61no—
243#121—61no—
244#120—62no—
245C4×Q8—62no—
246C4×D4—62no—
247C4×D4—62no—
248C4×D4—62no—
249C4×D4—62no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—62no—
253#117—32no—
254#117—62no—
255#117—61no—
256#117—61no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4C4×D311yescharacteristic
261C2×C4⋊C4C4×D311yescharacteristic
262C2×C4⋊C4D1211yescharacteristic
263C2×C4⋊C4D1211yescharacteristic
264C2×C42—22no—
265C2×C42—42no—
266#104—124no—
267C22⋊C8—62no—
268C22⋊C8—62no—
269#95—24no—
270#95—22no—
271#95—42no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3C2×C22⋊C411yescharacteristic
279C23×C3#12011yescharacteristic
280C23×C3#12011yescharacteristic
281C22×D3—22no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—81no—
311C2×C3×C4—81no—
312C2×C3×C4—81no—
313C2×C3×C4—81no—
314C2×C3×C4—81no—
315C2×C3×C4#10412yescomplemented
316C2×C3×C4C2×C22⋊C412yes—
317C2×C3×C4#12012yes—
318#67—42no—
319#67—42no—
320#67—42no—
321#67—42no—
322#67—42no—
323#67—42no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327D12—42no—
328D12—42no—
329D12—81no—
330C4×D3—44no—
331C4×D3—82no—
332C4×D3—82no—
333Dic6—42no—
334Dic6—42no—
335Dic6—81no—
336C3⋊C8—82no—
337C24C2×#6711yescharacteristic
338#41—122no—
339#41—122no—
340#41—122no—
341#41—122no—
342#41—241no—
343#41—241no—
344C2×Q8—32no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×D4—32no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4(C2×D3)⋊C411yescharacteristic
381C22×C4(C2×D3)⋊C411yescharacteristic
382C22×C4(C2×D3)⋊C411yescharacteristic
383C22×C4(C2×D3)⋊C411yescharacteristic
384C22×C4C2×C4×D311yescharacteristic
385C22×C4C2×D1211yescharacteristic
386C22×C4D3×D412yes—
387C22×C4D3×D412yes—
388M16—122no—
389M16—122no—
390C2×C8—122no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—32no—
396C4⋊C4—32no—
397C4⋊C4—61no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C22⋊C4—62no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22⋊C4—62no—
411C22⋊C4—62no—
412C22⋊C4—62no—
413C22⋊C4—62no—
414C22⋊C4—62no—
415C22⋊C4—62no—
416C42—24no—
417C42—42no—
418C42—62no—
419C42—62no—
420C42—62no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3#37911yescharacteristic
436C22×C3C2×#10411yescharacteristic
437C22×C3#42111yescharacteristic
438C22×C3#45411yescharacteristic
439C22×C3#45611yescharacteristic
440C22×C3B(4,𝔽2)11yescharacteristic
441C22×C3#45811yescharacteristic
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—42no—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—44no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—82no—
453Dic3—42no—
454Dic3—42no—
455Dic3—42no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—62no—
463C23C2×(C2×D3)⋊C411yescharacteristic
464C23#93711yescharacteristic
465C23#93711yescharacteristic
466Q8—122no—
467Q8—122no—
468Q8—241no—
469D4—122no—
470D4—122no—
471D4—122no—
472D4—122no—
473D4—122no—
474D4—122no—
475D4—122no—
476D4—122no—
477D4—241no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—124no—
515C2×C4—242no—
516C2×C4—242no—
517C2×C4#80512yescomplemented
518C2×C4D3×C22⋊C412yes—
519C2×C4#88212yes—
520C8—242no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3#178911yescharacteristic
528C2×C3#179011yescharacteristic
529C2×C3#179311yescharacteristic
530D3—82no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22C2×#80511yescharacteristic
549C22#495511yescharacteristic
550C22#500511yescharacteristic
551C22#503211yescharacteristic
552C22#505311yescharacteristic
553C22#506211yescharacteristic
554C22#509511yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—44no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—82no—
563C4—122no—
564C4—122no—
565C4—122no—
566C3#834711yescharacteristic, complemented, Sylow
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—242no—
574C2#6945411yescharacteristic
575C2#6959811yescharacteristic
576C2#6969211yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1436C2
socleC22×C3#379
prime radical#85718C2
cosocleC23×C3×C4C23
hypercenter#1436D3
hypocenterC3#8347
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1436

Complements

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

NH ≅ G/Ntype
2. C3×#1436C2semidirect
3. #71552C2semidirect
4. #71532C2semidirect
5. #69398C2semidirect
13. #5515C4semidirect
17. #5510C4semidirect
20. #5500C4semidirect
35. #1436D3semidirect
315. C2×C3×C4#104semidirect
517. C2×C4#805semidirect
566. C3#8347semidirect

Maximal subgroups

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

subgroupindex
#857182
C3×#14362
#715522
#715322
#693982
#83473