Finite Group Databasebeta

← #10691244

Subgroups — #10691244

Order 768.

516 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 105 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#73953C211yesmaximal, characteristic
2#73867C211yesmaximal, characteristic
3#69946C211yesmaximal, characteristic
4C2×#4469C211yesmaximal, characteristic
5#69510C211yesmaximal, characteristic
6C3×#1289C211yesmaximal, characteristic
7#67625C211yesmaximal, characteristic
8#9202—31nomaximal, Sylow
9C2×#924C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5212C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#4981—22no—
17#4978—22no—
18#4950—21no—
19#4943—21no—
20#4943—21no—
21Dic3×C22⋊C4—21no—
22#4659C412yescomplemented
23#4656C412yescomplemented
24#4655—22no—
25#4469—22no—
26#4467—22no—
27#4452—22no—
28#2627—31no—
29#2313—31no—
30#1520—31no—
31#1520—31no—
32C2×#324—31no—
33#1289—31no—
34#1289D311yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×Dic3D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×Dic6C2×C411yescharacteristic
39C22×C3×C8—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×#217—21no—
45C2×#217—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49#924—21no—
50#924—21no—
51#924—21no—
52#924—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4C2×C412yes—
59C2×Dic3⋊C4D412yes—
60C2×C4×Dic3—22no—
61C22×C3⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#95C2×C412yes—
65C3×#95D412yes—
66C3⋊#95—22no—
67C3⋊#95—24no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95D412yes—
72#830—21no—
73#830—21no—
74C2×C22⋊Q8—31no—
75C22×C4⋊C4—31no—
76C2×C22⋊C8—31no—
77C2×C22⋊C8C2×D311yescharacteristic
78#400—32no—
79#398—62no—
80#395—32no—
81#387—62no—
82#387—62no—
83#385—61no—
84#381—61no—
85#381—61no—
86#380—31no—
87#380C2×D311yescharacteristic
88#380C2×D311yescharacteristic
89C4×C22⋊C4—61no—
90#328—62no—
91#328—62no—
92#324—62no—
93C24×C3C2×D411yescharacteristic
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—81no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—81no—
115C22×Dic3C22⋊C412yes—
116C22×Dic3C22⋊C412yes—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—81no—
120C2×C3×C8—41no—
121C2×C3×C8—81no—
122C3×C4⋊C4—81no—
123C3×C4⋊C4—81no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128#217—41no—
129#217—41no—
130#217—41no—
131#217—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—82no—
135C4⋊Dic3—81no—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139Dic3⋊C4—22no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—82no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—81no—
145C22×Q8—31no—
146C23×C4—31no—
147C23×C4—31no—
148C23×C4C22×D311yescharacteristic
149C22×C8—21no—
150C22×C8—61no—
151C22⋊Q8—61no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—32no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—62no—
166C2×C4⋊C4—61no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C42—62no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174C22⋊C8—61no—
175C22⋊C8—61no—
176#95—32no—
177#95—62no—
178#95—62no—
179#95—62no—
180#95—62no—
181#95—64no—
182#95C4×D312yes—
183#95D1212yes—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3C2×C22⋊C411yescharacteristic
187C23×C3#12011yescharacteristic
188C23×C3#12311yescharacteristic
189C23×C3#13511yescharacteristic
190C23×C3#13511yescharacteristic
191C23×C3#13711yescharacteristic
192C23×C3#13711yescharacteristic
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—81no—
227Dic6—81no—
228Dic6—81no—
229C3×C8—81no—
230C3⋊C8—81no—
231C24D3×D411yescharacteristic
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—241no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—32no—
238C22×C4—32no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—62no—
250C22×C4—81no—
251C22×C4—241no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4C2×D1211yescharacteristic
254C22×C4#23511yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4#23911yescharacteristic
258C2×C8—41no—
259C2×C8—81no—
260C2×C8—121no—
261C2×C8—241no—
262C4⋊C4—62no—
263C4⋊C4—62no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—241no—
273C4⋊C4—242no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C42—242no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3#38611yescharacteristic
294C22×C3#41611yescharacteristic
295C22×C3#41911yescharacteristic
296C22×C3#45211yescharacteristic
297C22×C3#45211yescharacteristic
298C22×C3#48511yescharacteristic
299C22×C3#48511yescharacteristic
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—82no—
305C3×C4—82no—
306Dic3—42no—
307Dic3—42no—
308Dic3—82no—
309Dic3—81no—
310Dic3—81no—
311C23—41no—
312C23—41no—
313C23D3×C22⋊C411yescharacteristic
314C23#88211yescharacteristic
315C23#89411yescharacteristic
316C23#91211yescharacteristic
317C23#91511yescharacteristic
318C23#91611yescharacteristic
319C23#91811yescharacteristic
320Q8—241no—
321Q8—241no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—241no—
352C2×C4C4×D1211yescharacteristic
353C2×C4#88411yescharacteristic
354C2×C4#89311yescharacteristic
355C2×C4#89511yescharacteristic
356C8—81no—
357C8—241no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#187311yescharacteristic
365C2×C3#191211yescharacteristic
366C2×C3#191311yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22#466511yescharacteristic
378C22#477211yescharacteristic
379C22#478311yescharacteristic
380C22#480211yescharacteristic
381C22#480811yescharacteristic
382C22#486811yescharacteristic
383C22#488811yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—82no—
390C4—122no—
391C4—122no—
392C4—242no—
393C4—241no—
394C4—241no—
395C3#920211yescharacteristic, complemented, Sylow
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#6870811yescharacteristic
403C2#6887911yescharacteristic
404C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9202
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
22. #4659C4semidirect
23. #4656C4semidirect
395. C3#9202semidirect

Maximal subgroups

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

subgroupindex
#739532
#738672
#699462
C2×#44692
#695102
C3×#12892
#676252
#92023