Finite Group Databasebeta

← #10690158

Subgroups — #10690158

Order 768.

782 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 95 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#84095C211yesmaximal, characteristic
2#78634C211yesmaximal, characteristic, complemented
3C3×#1789C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5C4×#806C211yesmaximal, characteristic, complemented
6#67683C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9010—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C4×#67—21no—
11C2×C4×#67C2211yescharacteristic
12C2×C3×#104—22no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#837—22no—
16#5231C411yescharacteristic, complemented
17#5230C411yescharacteristic, complemented
18#5223—21no—
19#5223C411yescharacteristic, complemented
20#5222—21no—
21#5222C411yescharacteristic, complemented
22C4×#217—21no—
23C4×#217C2211yescharacteristic
24C4×#217C2211yescharacteristic
25#5207C2211yescharacteristic
26#5128—22no—
27C2×#806—21no—
28#4729—22no—
29#4726—21no—
30#4718—21no—
31#4527—21no—
32#2975—31no—
33#2388—31no—
34#1789—31no—
35#1789D311yescharacteristic, complemented
36C4×#99—31no—
37#1297—31no—
38#1297—31no—
39C2×C3×#41—21no—
40C2×C3×#41—21no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—44no—
44C22×C3×Q8C2×C411yescharacteristic
45C22×C3×D4—21no—
46C22×C3×D4C2×C411yescharacteristic
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×#67—21no—
50C22×#67D411yescharacteristic
51C22×C4×D3D411yescharacteristic
52C2×C3×M16D412yescomplemented
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59C2×#217—21no—
60C2×#217D411yescharacteristic
61C4×#67—21no—
62C4×#67—21no—
63C4×#67—21no—
64C4×#67—21no—
65C4×#67—44no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4C2×C411yescharacteristic
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4C2×C411yescharacteristic
70C2×C4×Dic3—21no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3D411yescharacteristic
73C2×#208D412yescomplemented
74C2×ΓL(1,𝔽25)—22no—
75C3×#104—44no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79#837—44no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95C2×C411yescharacteristic
84C3⋊#95C2×C411yescharacteristic
85#830—21no—
86#830—21no—
87(C2×D3)⋊C8—22no—
88Dic3⋊C8—22no—
89C8×Dic3—22no—
90#806—24no—
91C22×#41C2×D311yescharacteristic
92C2×C4×D4—31no—
93C2×C4×D4—61no—
94#436—62no—
95#436—62no—
96#431—61no—
97C2×#104—22no—
98C2×#104—62no—
99C2×#99—61no—
100#407—31no—
101#407—61no—
102#407C2×D311yescharacteristic
103#395—31no—
104#394—31no—
105#394—61no—
106#393—31no—
107#392—31no—
108#392—61no—
109C4×C22⋊C4—31no—
110C4×C22⋊C4—31no—
111C4×C22⋊C4—61no—
112C4×C22⋊C4C2×D311yescharacteristic
113#342—61no—
114C24×C3—21no—
115C24×C3C2×D411yescharacteristic
116C23×D3C22⋊C411yescharacteristic
117C3×#41—24no—
118C3×#41—24no—
119C3×#41—42no—
120C3×#41—42no—
121C3×#41—42no—
122C3×#41—42no—
123C3×#41—42no—
124C3×#41—42no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8—42no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4—42no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C22×C3×C4—24no—
136C22×C3×C4—21no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4C22×C411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C2×#67—21no—
149C2×#67—21no—
150C2×#67—22no—
151C2×#67—41no—
152C2×#67—42no—
153C2×#67—42no—
154C2×#67—42no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—41no—
158C22×Dic3C22⋊C411yescharacteristic
159C22×Dic3C22⋊C411yescharacteristic
160C22×Dic3C22⋊C411yescharacteristic
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—42no—
164C3×M16—42no—
165C2×C3×C8—22no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—41no—
168C3×C42—42no—
169#217—21no—
170#217—21no—
171#217—22no—
172#217—42no—
173#217—42no—
174#217—42no—
175#217—41no—
176#217—42no—
177(C2×D3)⋊C4—21no—
178(C2×D3)⋊C4—21no—
179(C2×D3)⋊C4—42no—
180Dic3⋊C4—21no—
181Dic3⋊C4—21no—
182Dic3⋊C4—42no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C4×Dic3—42no—
186C4×Dic3—42no—
187#208—42no—
188C2×C3⋊C8—22no—
189ΓL(1,𝔽25)—82no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—44no—
195C22×Q8C4×D311yescharacteristic
196C22×D4—21no—
197C22×D4—31no—
198C22×D4—61no—
199C22×D4C4×D311yescharacteristic
200C23×C4—21no—
201C23×C4—31no—
202C23×C4C22×D311yescharacteristic
203C2×M16—32no—
204C2×M16—62no—
205C2×M16#6712yescomplemented
206C4×D4—61no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4—61no—
210C4×D4—124no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—61no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—31no—
215C2×C22⋊C4—31no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C42—21no—
223C2×C42—31no—
224C2×C42—61no—
225C2×C42—61no—
226C4⋊C8—62no—
227#104—44no—
228#104—124no—
229#99—64no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—62no—
233C22⋊C8—61no—
234C22⋊C8—61no—
235C4×C8—62no—
236#95—21no—
237#95—31no—
238#95—31no—
239#95—61no—
240#95—61no—
241#95—61no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—22no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3C2×C22⋊C411yescharacteristic
254C23×C3#12111yescharacteristic
255C23×C3#12111yescharacteristic
256C22×D3—21no—
257C22×D3—21no—
258C22×D3—21no—
259C22×D3—41no—
260C22×D3—41no—
261C3×Q8—22no—
262C3×Q8—22no—
263C3×Q8—41no—
264C3×Q8—41no—
265C3×D4—22no—
266C3×D4—22no—
267C3×D4—41no—
268C3×D4—41no—
269C3×D4—44no—
270C3×D4—44no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—44no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—81no—
284C2×C3×C4—81no—
285C2×C3×C4C4×D411yescharacteristic
286C2×C3×C4C4×D411yescharacteristic
287C2×C3×C4#12011yescharacteristic
288C2×C3×C4#12311yescharacteristic
289#67—42no—
290#67—42no—
291#67—48no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—22no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307C2×Dic3#10412yescomplemented
308C4×D3—42no—
309C4×D3—81no—
310C3×C8—42no—
311C3⋊C8—42no—
312C24—21no—
313C24—31no—
314C24D3×D411yescharacteristic
315#41—24no—
316#41—24no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—42no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—62no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—121no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—122no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—122no—
355C22×C4—121no—
356C22×C4C2×C4×D311yescharacteristic
357C22×C4#23511yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4D3×D411yescharacteristic
360C22×C4#23711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362M16—42no—
363M16—122no—
364M16—242no—
365C2×C8—22no—
366C2×C8—62no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—122no—
370C22⋊C4—22no—
371C22⋊C4—41no—
372C22⋊C4—61no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—62no—
377C22⋊C4—122no—
378C22⋊C4—122no—
379C22⋊C4—122no—
380C22⋊C4—122no—
381C22⋊C4—122no—
382C22⋊C4—121no—
383C42—42no—
384C42—61no—
385C42—61no—
386C42—122no—
387C42—122no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3#38611yescharacteristic
406C22×C3C2×#10411yescharacteristic
407C22×C3#42111yescharacteristic
408C2×D3—42no—
409C2×D3—41no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—81no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—42no—
417C3×C4—82no—
418C3×C4B(4,𝔽2)12yes—
419Dic3—22no—
420Dic3—42no—
421Dic3—41no—
422Dic3—41no—
423Dic3—81no—
424Dic3—82no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—121no—
440C23—121no—
441C23D3×C22⋊C411yescharacteristic
442C23#88411yescharacteristic
443C23#93811yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—44no—
453D4—44no—
454D4—122no—
455D4—122no—
456D4—128no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—32no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—62no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4#67⋊C411yescharacteristic
490C2×C4#88311yescharacteristic
491C2×C4C4×#6711yescharacteristic
492C2×C4#93711yescharacteristic
493C8—42no—
494C8—122no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—82no—
500C2×C3#187111yescharacteristic
501C2×C3#188411yescharacteristic
502C2×C3#188911yescharacteristic
503D3—81no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—122no—
522C22—121no—
523C22—121no—
524C22—121no—
525C22—241no—
526C22D3×#10411yescharacteristic
527C22#482111yescharacteristic
528C22#495611yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—62no—
534C4—82no—
535C4—122no—
536C4—121no—
537C4—121no—
538C4—241no—
539C4—242no—
540C4#503212yes—
541C3#901011yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—41no—
545C2—41no—
546C2—82no—
547C2—241no—
548C2#6949911yescharacteristic
549C2#7038111yescharacteristic
550C2#7041711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#386
prime radical#84095C2
cosocleC23×C3×C4C23
hypercenter#1789D3
hypocenterC3#9010
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #78634C2semidirect
3. C3×#1789C2semidirect
4. #72138C2semidirect
5. C4×#806C2semidirect
6. #67683C2semidirect
7. #67618C2semidirect
16. #5231C4semidirect
17. #5230C4semidirect
19. #5223C4semidirect
21. #5222C4semidirect
35. #1789D3semidirect
52. C2×C3×M16D4semidirect
73. C2×#208D4semidirect
205. C2×M16#67semidirect
307. C2×Dic3#104semidirect
541. C3#9010semidirect

Maximal subgroups

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

subgroupindex
#840952
#786342
C3×#17892
#721382
C4×#8062
#676832
#676182
#90103