Finite Group Databasebeta

← #10711982

Subgroups — #10711982

Order 768.

672 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 89 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#74679C211yesmaximal, characteristic
2#70070C211yesmaximal, characteristic
3#70021C211yesmaximal, characteristic, complemented
4C2×#4474C211yesmaximal, characteristic
5#69597C211yesmaximal, characteristic, complemented
6C3×#1390C211yesmaximal, characteristic, complemented
7#67698C211yesmaximal, characteristic, complemented
8#12964—31nomaximal, Sylow
9C2×#940C2211yescharacteristic
10#5798C2211yescharacteristic
11C2×C4×#67—21no—
12C2×#67⋊C4—21no—
13C2×C3×#100C2211yescharacteristic
14C2×C3×#99C2211yescharacteristic
15C3×#386C2211yescharacteristic
16#5220C2211yescharacteristic
17C2×#834C2211yescharacteristic
18#5210—21no—
19#5131—21no—
20#4957—21no—
21#4957—21no—
22#4950—21no—
23#4665—21no—
24#4649—21no—
25#4555—21no—
26#4527—21no—
27#4527—21no—
28#4474C414yescomplemented
29#2366—31no—
30C2×#353—31no—
31#1887—31no—
32#1867—31no—
33#1867—31no—
34#1390—31no—
35#1390D311yescharacteristic, complemented
36C22×C3×D4C2311yescharacteristic
37C23×C3×C4D411yescharacteristic
38C22×#67—21no—
39C22×#67—21no—
40C23×Dic3D411yescharacteristic
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4D411yescharacteristic
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217D411yescharacteristic
52#940—21no—
53#940—21no—
54#940C2×C412yes—
55#940Q812yes—
56D4×Dic3—41no—
57D4×Dic3—41no—
58C4×#67—44no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62#926—21no—
63C2×C4⋊Dic3—41no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—41no—
67C2×C4×Dic3—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3D411yescharacteristic
70#67⋊C4—44no—
71C3×#100—21no—
72C3×#100—21no—
73C3×#99—21no—
74C3×#99C2×C412yescomplemented
75C3×#95—21no—
76#834—21no—
77#834C2×C412yescomplemented
78C3⋊#95—21no—
79C3⋊#95—41no—
80C3⋊#95—41no—
81#821—41no—
82#821—41no—
83C2×#127—31no—
84#518—31no—
85C2×C4×D4—61no—
86C2×C4×D4—61no—
87#418—61no—
88C2×#100C2×D311yescharacteristic
89C2×#99—31no—
90C2×#99C2×D311yescharacteristic
91#390—61no—
92#390—61no—
93#386—31no—
94#386—61no—
95#386C2×D311yescharacteristic
96#385—61no—
97#385—61no—
98C4×C4⋊C4—61no—
99#353—34no—
100#342—61no—
101#342—61no—
102#339—61no—
103C24×C3C2×D411yescharacteristic
104C24×C3C2×D411yescharacteristic
105C23×D3—21no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4C22×C411yescharacteristic
109C2×C3×D4C2×Q811yescharacteristic
110C2×C3×D4#4111yescharacteristic
111C2×C3×D4#4111yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C2×#67—22no—
119C2×#67—22no—
120C2×#67—22no—
121C2×#67—22no—
122C2×#67—42no—
123C2×#67—42no—
124C2×#67—42no—
125C2×#67—42no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—42no—
134C22×Dic3—41no—
135C22×Dic3—41no—
136C2×D12—21no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12C4⋊C412yescomplemented
140C2×C4×D3—42no—
141C2×C4×D3—42no—
142C2×C4×D3—81no—
143C3×M16—41no—
144C3×M16—41no—
145C2×C3×C8—41no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150#217—22no—
151#217—42no—
152#217—41no—
153#217—41no—
154#217—41no—
155#217—41no—
156#217—41no—
157(C2×D3)⋊C4—42no—
158(C2×D3)⋊C4—42no—
159(C2×D3)⋊C4—81no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—82no—
163Dic3⋊C4—24no—
164Dic3⋊C4—42no—
165Dic3⋊C4—82no—
166C4×Dic3—21no—
167C4×Dic3—42no—
168C4×Dic3—42no—
169C4×Dic3—41no—
170C4×Dic3C4⋊C412yescomplemented
171C22×D4—31no—
172C22×D4—61no—
173C22×D4—61no—
174C22×D4C22×D311yescharacteristic
175C23×C4—31no—
176C23×C4—61no—
177C23×C4D1211yescharacteristic
178C2×M16—21no—
179#127—32no—
180#127—32no—
181#127—61no—
182#127—61no—
183C4×D4—124no—
184C4×D4—124no—
185C4×D4—121no—
186C4×D4—121no—
187#117—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—121no—
191C2×C4⋊C4—121no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4D1211yescharacteristic
203C2×C42—31no—
204C2×C42—61no—
205C2×C42—61no—
206#102—121no—
207#102—121no—
208#100—21no—
209#100—21no—
210#99—21no—
211#99—32no—
212#99—61no—
213#99C4×D312yescomplemented
214#95—21no—
215#95—61no—
216#95—121no—
217#95—121no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3C4×D411yescharacteristic
227C23×C3C4×D411yescharacteristic
228C23×C3#12011yescharacteristic
229C23×C3C22⋊Q811yescharacteristic
230C23×C3C22⋊Q811yescharacteristic
231C22×D3—22no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4C2×C4⋊C411yescharacteristic
256C2×C3×C4#12311yescharacteristic
257#67—44no—
258#67—44no—
259#67—44no—
260#67—44no—
261C2×Dic3—22no—
262C2×Dic3—22no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—81no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C2×Dic3—81no—
281D12—42no—
282D12—81no—
283C4×D3—84no—
284C3×C8—81no—
285C24—61no—
286C24C2×D1211yescharacteristic
287C24D3×D411yescharacteristic
288C2×D4—32no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—61no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—122no—
299C2×D4—122no—
300C2×D4—122no—
301C2×D4—122no—
302C2×D4C2×C4×D311yescharacteristic
303C2×D4#23511yescharacteristic
304C2×D4#23711yescharacteristic
305C2×D4D3×Q811yescharacteristic
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—122no—
319C22×C4—122no—
320C22×C4—122no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4—241no—
324C22×C4D3×D411yescharacteristic
325M16—41no—
326M16—41no—
327C2×C8—41no—
328C4⋊C4—64no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—122no—
332C4⋊C4—242no—
333C4⋊C4—242no—
334C22⋊C4—22no—
335C22⋊C4—42no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—62no—
339C22⋊C4—122no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C22⋊C4—121no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—241no—
348C42—32no—
349C42—61no—
350C42—121no—
351C42—122no—
352C42—122no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—81no—
368C22×C3—81no—
369C22×C3#38511yescharacteristic
370C22×C3B(4,𝔽2)11yescharacteristic
371C22×C3#45811yescharacteristic
372C2×D3—42no—
373C2×D3—42no—
374C2×D3—42no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—81no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—42no—
382C3×C4—81no—
383C3×C4—82no—
384Dic3—24no—
385Dic3—42no—
386Dic3—81no—
387Dic3—82no—
388Dic3—81no—
389Dic3—81no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—62no—
399C23—121no—
400C23—121no—
401C23—121no—
402C23—121no—
403C23—121no—
404C23—121no—
405C23C4×D1211yescharacteristic
406C23#67⋊C411yescharacteristic
407C23#88211yescharacteristic
408C23#89611yescharacteristic
409C23#89711yescharacteristic
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—122no—
415D4—124no—
416D4—124no—
417D4—124no—
418D4—124no—
419D4—241no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—122no—
440C2×C4—121no—
441C2×C4—122no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—244no—
447C2×C4—241no—
448C2×C4—241no—
449C2×C4—241no—
450C2×C4—241no—
451C2×C4—241no—
452C2×C4—241no—
453C2×C4—241no—
454C2×C4#88311yescharacteristic
455C2×C4D3×C4⋊C411yescharacteristic
456C8—81no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3#186711yescharacteristic
464C2×C3#219611yescharacteristic
465C2×C3#219811yescharacteristic
466D3—82no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—81no—
482C22—81no—
483C22—122no—
484C22—122no—
485C22—122no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22—241no—
490C22#466811yescharacteristic
491C22#474111yescharacteristic
492C22#474211yescharacteristic
493C4—21no—
494C4—21no—
495C4—42no—
496C4—64no—
497C4—81no—
498C4—82no—
499C4—122no—
500C4—241no—
501C4—242no—
502C4—241no—
503C4—241no—
504C3#1296411yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—41no—
508C2—41no—
509C2—41no—
510C2—41no—
511C2—242no—
512C2#6896111yescharacteristic
513C2#6913011yescharacteristic
514C2#6914011yescharacteristic
subgroupG/H
centerC22#4741
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1390C2
socleC22×C3B(4,𝔽2)
prime radicalC2×#940C22
cosocleC22×C3×D4C23
hypercenter#1390D3
hypocenterC3#12964
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
3. #70021C2semidirect
5. #69597C2semidirect
6. C3×#1390C2semidirect
7. #67698C2semidirect
28. #4474C4semidirect
35. #1390D3semidirect
74. C3×#99C2×C4semidirect
77. #834C2×C4semidirect
139. C2×D12C4⋊C4semidirect
170. C4×Dic3C4⋊C4semidirect
213. #99C4×D3semidirect
504. C3#12964semidirect

Maximal subgroups

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

subgroupindex
#746792
#700702
#700212
C2×#44742
#695972
C3×#13902
#676982
#129643