Finite Group Databasebeta

← #10695906

Subgroups — #10695906

Order 768.

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

normal subgroups only

This group has 536 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#78642C211yesmaximal, characteristic
2C2×#4722C211yesmaximal, characteristic, complemented
3C2×#4736C211yesmaximal, characteristic
4#69973C211yesmaximal, characteristic, complemented
5#69923C211yesmaximal, characteristic, complemented
6#69921C211yesmaximal, characteristic
7C3×#1298C211yesmaximal, characteristic, complemented
8#9994—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15#5207C2211yescharacteristic
16#5134—22no—
17C2×#821—21no—
18C2×#818—21no—
19C2×#818—21no—
20C2×#816—21no—
21#4738—21no—
22#4736—21no—
23#4736—22no—
24#4736—22no—
25#4731—21no—
26#4722—21no—
27#4722—22no—
28#4722—22no—
29#4554—22no—
30#4469—22no—
31#4468—22no—
32#3193—31no—
33C2×#484—31no—
34C2×#480—31no—
35#1821—31no—
36#1472—31no—
37#1471—31no—
38#1298D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C22×D12D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×M16D412yes—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C2×#217—21no—
48#930—21no—
49#930—21no—
50#930—21no—
51#930—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D412yes—
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×#208—21no—
67C2×#204—22no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×C22⋊C8D412yes—
74#830—21no—
75#830—21no—
76#821—22no—
77#821—24no—
78#818—22no—
79#818—24no—
80#818—24no—
81#816—22no—
82#816—24no—
83C2×C22⋊Q8—31no—
84C2×#121—31no—
85C22×C4⋊C4—31no—
86#484—62no—
87#484—61no—
88#484—62no—
89#480—62no—
90#480—61no—
91#480—62no—
92#468—62no—
93#450—61no—
94#448—61no—
95C2×#107—61no—
96C2×#107—61no—
97C2×#103—61no—
98C2×#102—61no—
99#407—31no—
100#407C2×D311yescharacteristic
101C2×C22⋊C8C2×D311yescharacteristic
102C2×C22⋊C8C2×D311yescharacteristic
103#343—62no—
104#324—62no—
105#323—62no—
106C24×C3#4111yescharacteristic
107C23×D3—21no—
108C22×C3×C4—42no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—41no—
116C2×#67—41no—
117C2×#67—82no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—22no—
121C22×Dic3—21no—
122C22×Dic3—41no—
123C22×Dic3—41no—
124C22×Dic3—82no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—42no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—42no—
131C3×M16—42no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—42no—
136C2×C3×C8—42no—
137#217—41no—
138#217—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151#208—42no—
152C2×C3⋊C8—41no—
153#204—82no—
154C22×Q8—31no—
155C22×D4—31no—
156C22×D4—61no—
157C23×C4—61no—
158C23×C4C22×D311yescharacteristic
159C2×SD16—62no—
160C2×M16—61no—
161C2×M16D1212yes—
162C22×C8—21no—
163C22×C8—21no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171#121—61no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183#107—64no—
184#107—64no—
185#107—62no—
186#103—64no—
187#103—62no—
188#102—64no—
189#102—62no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196C22⋊C8D1212yes—
197C22⋊C8D1212yes—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—41no—
201C23×C3#12111yescharacteristic
202C23×C3#12111yescharacteristic
203C23×C3#12111yescharacteristic
204C22×D3—41no—
205C22×D3—42no—
206C22×D3—81no—
207C22×D3—81no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12711yescharacteristic
218#67—82no—
219#67—82no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225C2×Dic3—82no—
226C2×Dic3—82no—
227C2×Dic3—81no—
228C2×Dic3—82no—
229C2×Dic3—81no—
230D12—42no—
231D12—42no—
232Dic6—42no—
233Dic6—42no—
234C3×C8—42no—
235C3×C8—42no—
236C3×C8—42no—
237C3⋊C8—81no—
238C24—61no—
239C24#23911yescharacteristic
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—122no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—122no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—242no—
249C22×C4—42no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—62no—
254C22×C4—121no—
255C22×C4—121no—
256C22×C4—242no—
257C22×C4C2×D1211yescharacteristic
258C22×C4C2×D1211yescharacteristic
259C22×C4C2×D1211yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263SD16—242no—
264M16—42no—
265M16—122no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—121no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—62no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—44no—
294C22×C3#39211yescharacteristic
295C22×C3#47212yes—
296C22×C3#47312yes—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—162no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4#45812yes—
305Dic3—81no—
306Dic3—82no—
307Dic3—81no—
308Dic3—81no—
309C23—22no—
310C23—22no—
311C23—41no—
312C23—121no—
313C23—122no—
314C23—241no—
315C23—241no—
316C23#89511yescharacteristic
317C23#89511yescharacteristic
318C23#89511yescharacteristic
319Q8—122no—
320Q8—122no—
321D4—122no—
322D4—122no—
323D4—242no—
324D4—242no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—122no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—122no—
336C2×C4—242no—
337C2×C4—242no—
338C2×C4—242no—
339C2×C4—241no—
340C2×C4—241no—
341C2×C4#87411yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#88211yescharacteristic
344C2×C4#88211yescharacteristic
345C8—42no—
346C8—42no—
347C8—42no—
348C8—241no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—41no—
352C2×C3#201811yescharacteristic
353C2×C3#202011yescharacteristic
354C2×C3#202211yescharacteristic
355D3—161no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—44no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22—482no—
368C22#467111yescharacteristic
369C22#489612yes—
370C22#489712yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—241no—
375C4—242no—
376C4—241no—
377C4—241no—
378C4#474212yes—
379C3#999411yescharacteristic, complemented, Sylow
380C2—22no—
381C2—22no—
382C2—41no—
383C2—481no—
384C2#6894211yescharacteristic
385C2#6894411yescharacteristic
386C2#6896611yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1298C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1298D3
hypocenterC3#9994
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. C2×#4722C2semidirect
4. #69973C2semidirect
5. #69923C2semidirect
7. C3×#1298C2semidirect
38. #1298D3semidirect
379. C3#9994semidirect

Maximal subgroups

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

subgroupindex
#786422
C2×#47222
C2×#47362
#699732
#699232
#699212
C3×#12982
#99943