Finite Group Databasebeta

← #10698403

Subgroups — #10698403

Order 768.

438 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 75 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#74093C211yesmaximal, characteristic, complemented
2C3×#1510C211yesmaximal, characteristic
3#71686C211yesmaximal, characteristic
4#71673C211yesmaximal, characteristic
5#67685C211yesmaximal, characteristic, complemented
6#67683C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#10384—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C3×#407C2211yescharacteristic
11C3×#381C2211yescharacteristic
12C2×#834—21no—
13C2×#834—21no—
14#5212C2211yescharacteristic
15C4×#217C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18#4949—21no—
19#4948—21no—
20#4661—21no—
21#4657—21no—
22#4655—21no—
23#4647—21no—
24C3×#327—21no—
25C3×#327—21no—
26#4537—21no—
27#4537—21no—
28#2667—31no—
29#1964—31no—
30#1510D311yescharacteristic
31#1507—31no—
32#1305—31no—
33#1297—31no—
34#1284—31no—
35C22×C3×D4—21no—
36C22×C3×D4Q811yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C2×C3×M16—21no—
39C3×#121—21no—
40C3×#121—21no—
41C3×#121—21no—
42C3×#121—21no—
43C2×C3×C4⋊C4Q811yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×#217—21no—
47C2×#217—21no—
48C2×#217D411yescharacteristic
49C2×#217D411yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×#208—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95D411yescharacteristic
64C3×#95D411yescharacteristic
65#834—42no—
66#834—42no—
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—
78#407—31no—
79#407C2×D311yescharacteristic
80#399—61no—
81#398—61no—
82#398—61no—
83#396—61no—
84#389—61no—
85#387—61no—
86#381C2×D311yescharacteristic
87#380—31no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4—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—41no—
120C3×M16—81no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—81no—
127#217—22no—
128#217—42no—
129#217—41no—
130#217—41no—
131#217—41no—
132C4⋊Dic3—81no—
133C4⋊Dic3—81no—
134C4⋊Dic3—81no—
135Dic3⋊C4—42no—
136Dic3⋊C4—42no—
137Dic3⋊C4—81no—
138Dic3⋊C4—81no—
139C4×Dic3—81no—
140C4×Dic3—81no—
141#208—81no—
142C2×C3⋊C8—41no—
143C22×D4—21no—
144C22×D4Dic611yescharacteristic
145C23×C4C22×D311yescharacteristic
146C2×M16—21no—
147C2×M16—61no—
148#121—21no—
149#121—21no—
150#121—21no—
151#121—21no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4Dic611yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C42—61no—
167C2×C42—61no—
168#99—122no—
169#99—122no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C22⋊C8—61no—
173C22⋊C8—61no—
174#95—21no—
175#95—31no—
176#95—31no—
177#95—61no—
178#95—61no—
179#95—61no—
180#95—61no—
181#95#6711yescharacteristic
182#95#6711yescharacteristic
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—81no—
191C23×C3—81no—
192C23×C3#12011yescharacteristic
193C23×C3C22⋊Q811yescharacteristic
194C23×C3#12411yescharacteristic
195C3×D4—81no—
196C3×D4—81no—
197C3×D4—81no—
198C3×D4—81no—
199C3×D4—81no—
200C3×D4—81no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
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×C4C22⋊Q811yescharacteristic
217C2×C3×C4C22⋊Q811yescharacteristic
218C2×C3×C4#12411yescharacteristic
219C2×C3×C4#12411yescharacteristic
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C3×C8—81no—
236C3⋊C8—81no—
237C24—21no—
238C24D3×D411yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—81no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—81no—
256C22×C4—121no—
257C22×C4C2×Dic611yescharacteristic
258C22×C4#23511yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4#23711yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263M16—81no—
264M16—241no—
265C2×C8—41no—
266C2×C8—121no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—122no—
270C4⋊C4—122no—
271C4⋊C4—241no—
272C4⋊C4—241no—
273C4⋊C4—241no—
274C4⋊C4—241no—
275C4⋊C4—241no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—62no—
279C22⋊C4—81no—
280C22⋊C4—122no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C42—241no—
285C42—241no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—161no—
300C22×C3—161no—
301C22×C3#39311yescharacteristic
302C22×C3#45311yescharacteristic
303C22×C3#45411yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4—81no—
311Dic3—42no—
312Dic3—81no—
313Dic3—81no—
314Dic3—81no—
315Dic3—82no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—81no—
324C23—81no—
325C23#87811yescharacteristic
326C23#88511yescharacteristic
327C23#93711yescharacteristic
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—62no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—241no—
363C2×C4—241no—
364C2×C4#87811yescharacteristic
365C2×C4#88511yescharacteristic
366C2×C4#92411yescharacteristic
367C2×C4#93611yescharacteristic
368C8—81no—
369C8—241no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—161no—
375C2×C3#202711yescharacteristic
376C2×C3#203111yescharacteristic
377C2×C3#203211yescharacteristic
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—161no—
392C22—161no—
393C22#494711yescharacteristic
394C22#519811yescharacteristic
395C22#520111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—122no—
404C4—241no—
405C4—241no—
406C4—241no—
407C4—242no—
408C3#1038411yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—41no—
412C2—41no—
413C2—161no—
414C2#7002511yescharacteristic
415C2#7031311yescharacteristic
416C2#7039411yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#393
prime radicalC2×C3×#121C22
hypercenter#1510D3
hypocenterC3#10384
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
1. #74093C2semidirect
5. #67685C2semidirect
6. #67683C2semidirect
7. #67628C2semidirect
408. C3#10384semidirect

Maximal subgroups

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

subgroupindex
#740932
C3×#15102
#716862
#716732
#676852
#676832
#676282
#103843