Finite Group Databasebeta

← #10695903

Subgroups — #10695903

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
1C2×C3×#480C211yesmaximal, characteristic
2#82631C211yesmaximal, characteristic
3C2×#4970C211yesmaximal, characteristic, complemented
4#71676C211yesmaximal, characteristic
5#71562C211yesmaximal, characteristic, complemented
6#69923C211yesmaximal, characteristic, complemented
7#67680C211yesmaximal, characteristic, complemented
8#9994—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#480—22no—
13C3×#480—22no—
14C2×C3×#107—21no—
15C2×C3×#102—21no—
16C2×C3×C22⋊C8C2211yescharacteristic
17#5218—21no—
18#5207C2211yescharacteristic
19#5207C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#5042—22no—
22#5040—21no—
23#5033—21no—
24#4970—21no—
25#4970—22no—
26#4970—22no—
27C2×#810—21no—
28C2×#807—21no—
29#4530—22no—
30#4525—22no—
31#4469—22no—
32#3193—31no—
33C2×#484—31no—
34C2×#480D311yescharacteristic
35#1821—31no—
36#1472—31no—
37#1471—31no—
38#1298—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4—21no—
57C2×#217—21no—
58C2×#214—22no—
59C2×C4⋊Dic3D411yescharacteristic
60#924—21no—
61#924—21no—
62#924—21no—
63#924—21no—
64C2×Dic3⋊C4—21no—
65C2×#208—21no—
66C2×#208D412yes—
67C22×C3⋊C8—21no—
68C3×#107—24no—
69C3×#102—24no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D412yes—
72#832—22no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830D412yes—
79#810—24no—
80#810—22no—
81#807—24no—
82#807—22no—
83C2×C22⋊Q8—31no—
84C2×#121C2×D311yescharacteristic
85C22×C4⋊C4C2×D311yescharacteristic
86#484—62no—
87#484—62no—
88#484—61no—
89#480—22no—
90#480—22no—
91#480—61no—
92#468—62no—
93#450—61no—
94#448—61no—
95C2×#107—21no—
96C2×#107—61no—
97C2×#103—61no—
98C2×#102—21no—
99#407—31no—
100#407—31no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#343—62no—
104#324—62no—
105#323—62no—
106C24×C3—21no—
107C24×C3#4111yescharacteristic
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—42no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—82no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—42no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—82no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—42no—
131C2×C3×C8—22no—
132C2×C3×C8—42no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141#217—41no—
142#217—41no—
143#214—82no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148#208—42no—
149#208—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—42no—
153C2×C3⋊C8—41no—
154C22×Q8—31no—
155C22×D4—21no—
156C22×D4#6711yescharacteristic
157C23×C4—21no—
158C23×C4C22×D311yescharacteristic
159C2×SD16—62no—
160C2×M16—32no—
161C2×M16—61no—
162C22×C8—21no—
163C22×C8—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168#121—21no—
169#121—21no—
170#121—21no—
171#121—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4#6712yes—
180C2×C4⋊C4#6711yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—61no—
183#107—24no—
184#107—64no—
185#107—62no—
186#103—64no—
187#103—62no—
188#102—24no—
189#102—62no—
190C22⋊C8—21no—
191C22⋊C8—32no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C22⋊C8#6712yes—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3#12111yescharacteristic
206C23×C3#12111yescharacteristic
207C23×C3#12111yescharacteristic
208C3×D4—42no—
209C3×D4—42no—
210C3×D4—82no—
211C3×D4—82no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—82no—
221C2×C3×C4—82no—
222C2×C3×C4—82no—
223C2×C3×C4—81no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—81no—
232Dic6—42no—
233Dic6—42no—
234C3×C8—42no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237C3⋊C8—81no—
238C24—21no—
239C24#23711yescharacteristic
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—122no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—82no—
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263SD16—242no—
264M16—122no—
265M16—122no—
266C2×C8—22no—
267C2×C8—42no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—122no—
271C2×C8—121no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C22⋊C4—41no—
283C22⋊C4—41no—
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—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—162no—
298C22×C3#39211yescharacteristic
299C22×C3#47212yes—
300C22×C3#47312yes—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—82no—
305C3×C4—81no—
306C3×C4#45812yes—
307Dic3—81no—
308Dic3—81no—
309C23—22no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23#93811yescharacteristic
317C23#93911yescharacteristic
318C23#93911yescharacteristic
319Q8—122no—
320Q8—122no—
321D4—42no—
322D4—42no—
323D4—82no—
324D4—82no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—122no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—241no—
341C2×C4#93711yescharacteristic
342C2×C4#93711yescharacteristic
343C2×C4#94011yescharacteristic
344C2×C4#95311yescharacteristic
345C8—42no—
346C8—122no—
347C8—122no—
348C8—241no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—41no—
352C2×C3—161no—
353C2×C3#201811yescharacteristic
354C2×C3#202011yescharacteristic
355C2×C3#202211yescharacteristic
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—44no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—162no—
368C22#516012yes—
369C22#520412yes—
370C22#522311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—81no—
376C4—241no—
377C4—241no—
378C4#505312yes—
379C3#999411yescharacteristic, complemented, Sylow
380C2—22no—
381C2—22no—
382C2—41no—
383C2—161no—
384C2#7170611yescharacteristic
385C2#7204611yescharacteristic
386C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#480C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenterC2×#480D3
hypocenterC3#9994
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
3. C2×#4970C2semidirect
5. #71562C2semidirect
6. #69923C2semidirect
7. #67680C2semidirect
379. C3#9994semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4802
#826312
C2×#49702
#716762
#715622
#699232
#676802
#99943