Finite Group Databasebeta

← #10698347

Subgroups — #10698347

Order 768.

464 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 91 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#73945C211yesmaximal, characteristic, complemented
2C3×#1519C211yesmaximal, characteristic
3#71686C211yesmaximal, characteristic
4C2×#4537C211yesmaximal, characteristic
5#67677C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10376—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×C22⋊C8C2211yescharacteristic
11C3×#380C2211yescharacteristic
12C2×#834—21no—
13C2×#834—21no—
14#5212C2211yescharacteristic
15C4×#217C2211yescharacteristic
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4986—21no—
19#4982—21no—
20#4950—21no—
21#4942—21no—
22#4655—21no—
23#4654—21no—
24C3×#327—21no—
25C3×#327—21no—
26#4537—21no—
27#4537—21no—
28#2569—31no—
29#1964—31no—
30#1519D311yescharacteristic
31C2×#327—31no—
32#1294—31no—
33#1289—31no—
34#1282—31no—
35C22×C3×D4—21no—
36C22×C3×D4Q811yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—21no—
39C22×C3×C8—21no—
40C3×#121—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×#121—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4Q811yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×#217—21no—
49C2×#217D411yescharacteristic
50C2×#217D411yescharacteristic
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—21no—
57C22×C3⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×#95D411yescharacteristic
61C3×#95D411yescharacteristic
62#834—42no—
63#834—42no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#830—21no—
74#830—21no—
75C2×#121C2×D311yescharacteristic
76C2×#99—61no—
77C2×#99—61no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8C2×D311yescharacteristic
80#400—61no—
81#398—61no—
82#387—61no—
83#387—61no—
84#385—61no—
85#380—31no—
86#380—61no—
87#380C2×D311yescharacteristic
88C4×C22⋊C4—31no—
89C2×#95—31no—
90#327—21no—
91#327—21no—
92#327—61no—
93#327—61no—
94C24×C3—21no—
95C24×C3C2×D411yescharacteristic
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—81no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—81no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×Q811yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—42no—
120C22×Dic3—41no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C22×Dic3—81no—
124C2×C3×C8—41no—
125C2×C3×C8—81no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—81no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133#217—21no—
134#217—21no—
135#217—21no—
136#217—21no—
137#217—41no—
138#217—41no—
139C4⋊Dic3—42no—
140C4⋊Dic3—81no—
141C4⋊Dic3—81no—
142Dic3⋊C4—42no—
143Dic3⋊C4—81no—
144C4×Dic3—81no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—81no—
147C22×D4—21no—
148C22×D4Dic611yescharacteristic
149C23×C4—61no—
150C23×C4C22×D311yescharacteristic
151C22×C8—21no—
152C22×C8—61no—
153#121—21no—
154#121—21no—
155#121—21no—
156#121—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4Dic611yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—61no—
169C2×C42—61no—
170#99—122no—
171#99—122no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174C22⋊C8—61no—
175C22⋊C8—61no—
176#95—31no—
177#95—31no—
178#95—61no—
179#95—61no—
180#95—61no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95#6711yescharacteristic
186#95#6711yescharacteristic
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—81no—
193C23×C3—81no—
194C23×C3#12011yescharacteristic
195C23×C3C22⋊Q811yescharacteristic
196C23×C3#12411yescharacteristic
197C23×C3#13511yescharacteristic
198C23×C3#13511yescharacteristic
199C23×C3AGL(1,ℤ/8)11yescharacteristic
200C23×C3AGL(1,ℤ/8)11yescharacteristic
201C3×D4—81no—
202C3×D4—81no—
203C3×D4—81no—
204C3×D4—81no—
205C3×D4—81no—
206C3×D4—81no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4C22⋊Q811yescharacteristic
223C2×C3×C4C22⋊Q811yescharacteristic
224C2×C3×C4#12411yescharacteristic
225C2×C3×C4#12411yescharacteristic
226C2×Dic3—22no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—82no—
239C2×Dic3—82no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C3×C8—81no—
244C3⋊C8—81no—
245C24—21no—
246C24D3×D411yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—81no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—81no—
264C22×C4—122no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—241no—
269C22×C4C2×Dic611yescharacteristic
270C22×C4#23511yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4#23711yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275C2×C8—41no—
276C2×C8—81no—
277C2×C8—121no—
278C2×C8—241no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—81no—
282C4⋊C4—122no—
283C4⋊C4—122no—
284C4⋊C4—241no—
285C4⋊C4—241no—
286C4⋊C4—241no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C42—241no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—161no—
312C22×C3—161no—
313C22×C3#39311yescharacteristic
314C22×C3#44911yescharacteristic
315C22×C3#44911yescharacteristic
316C22×C3#47811yescharacteristic
317C22×C3#47811yescharacteristic
318C22×C3#48811yescharacteristic
319C22×C3#49011yescharacteristic
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—81no—
325C3×C4—81no—
326C3×C4—81no—
327Dic3—42no—
328Dic3—81no—
329Dic3—81no—
330Dic3—81no—
331Dic3—161no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—81no—
338C23—81no—
339C23#87811yescharacteristic
340C23#88511yescharacteristic
341C23#91111yescharacteristic
342C23#91211yescharacteristic
343C23#91411yescharacteristic
344C23#91611yescharacteristic
345C23#93711yescharacteristic
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—62no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—242no—
380C2×C4—242no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4#87811yescharacteristic
385C2×C4#88511yescharacteristic
386C2×C4#92411yescharacteristic
387C2×C4#93611yescharacteristic
388C8—81no—
389C8—241no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—161no—
397C2×C3#202511yescharacteristic
398C2×C3#203111yescharacteristic
399C2×C3#203211yescharacteristic
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—161no—
414C22—161no—
415C22#476311yescharacteristic
416C22#476411yescharacteristic
417C22#476611yescharacteristic
418C22#476811yescharacteristic
419C22#494711yescharacteristic
420C22#515811yescharacteristic
421C22#517211yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—122no—
430C4—241no—
431C4—241no—
432C4—241no—
433C4—481no—
434C3#1037611yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—161no—
442C2#7002511yescharacteristic
443C2#7016411yescharacteristic
444C2#7039411yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#393
prime radicalC2×C3×#121C22
hypercenter#1519D3
hypocenterC3#10376
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
1. #73945C2semidirect
5. #67677C2semidirect
6. #67675C2semidirect
7. #67625C2semidirect
434. C3#10376semidirect

Maximal subgroups

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

subgroupindex
#739452
C3×#15192
#716862
C2×#45372
#676772
#676752
#676252
#103763