Finite Group Databasebeta

← #10695847

Subgroups — #10695847

Order 768.

502 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#3095C211yesmaximal, characteristic
2#82808C211yesmaximal, characteristic
3#82220C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic
5#71594C211yesmaximal, characteristic
6#69981C211yesmaximal, characteristic
7#67686C211yesmaximal, characteristic
8#9985—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#485—22no—
13C3×#484—21no—
14C3×#483—21no—
15C3×#428—21no—
16C3×#419—21no—
17C3×#407C2211yescharacteristic
18#5230—21no—
19#5211C2211yescharacteristic
20#5211C2211yescharacteristic
21#5207C2211yescharacteristic
22#5004—21no—
23#4996—22no—
24#4995—21no—
25#4986—21no—
26#4985—21no—
27#4979—21no—
28#4971—21no—
29#4970—21no—
30#4952—22no—
31#4538—22no—
32#4526—22no—
33#4467—22no—
34#3095—31no—
35#3095D311yescharacteristic
36#2817—31no—
37#1820—31no—
38#1511—31no—
39#1511—31no—
40#1306—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×#117—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217D412yes—
63C2×C4⋊Dic3D411yescharacteristic
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—22no—
70C2×#208—21no—
71C3×#107—22no—
72C3×#106—22no—
73C3×#103—22no—
74C3×#103—22no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yes—
86#810—22no—
87#810—22no—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93#485—22no—
94#485—62no—
95#484—21no—
96#484—61no—
97#483—21no—
98#483—61no—
99#428—21no—
100#428—61no—
101#419—21no—
102#419—61no—
103#407—31no—
104#407C2×D311yescharacteristic
105#400—61no—
106#400—61no—
107#397—62no—
108#395—61no—
109#395—61no—
110#381—31no—
111#381—31no—
112#342—62no—
113#328—62no—
114#328—62no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—42no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×Dic3—22no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—41no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—42no—
137C3×M16—42no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42—41no—
152C3×C42—41no—
153#217—41no—
154#217—41no—
155#217—42no—
156#217—42no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161Dic3⋊C4—82no—
162#208—42no—
163C2×C3⋊C8—22no—
164C22×Q8—31no—
165C22×Q8#6711yescharacteristic
166C23×C4C22×D311yescharacteristic
167C2×M16—21no—
168C2×M16—61no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177#117—21no—
178#117—21no—
179#117—21no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C4⋊C4#6711yescharacteristic
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194#107—22no—
195#107—62no—
196#106—22no—
197#106—62no—
198#103—22no—
199#103—22no—
200#103—62no—
201#103—62no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205C22⋊C8#6712yes—
206#95—62no—
207#95—62no—
208#95—62no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3#12011yescharacteristic
217C23×C3#12111yescharacteristic
218C23×C3#12711yescharacteristic
219C3×Q8—42no—
220C3×Q8—81no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249Dic6—42no—
250Dic6—81no—
251C3×C8—42no—
252C3⋊C8—42no—
253C24D3×D411yescharacteristic
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—42no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—122no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—62no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—121no—
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271C22×C4#23711yescharacteristic
272C22×C4C2×#6711yescharacteristic
273C22×C4C2×#6711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275M16—42no—
276M16—122no—
277C2×C8—22no—
278C2×C8—62no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—242no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—122no—
299C22⋊C4—122no—
300C42—41no—
301C42—41no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—82no—
311C22×C3#39211yescharacteristic
312C22×C3#45812yes—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—81no—
318C3×C4—81no—
319C3×C4—81no—
320C3×C4—81no—
321Dic3—82no—
322Dic3—81no—
323Dic3—81no—
324Dic3—161no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23#93711yescharacteristic
329C23#93911yescharacteristic
330C23#94011yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—122no—
334Q8—241no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—241no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4#93711yescharacteristic
360C2×C4#93811yescharacteristic
361C2×C4#93911yescharacteristic
362C2×C4#95311yescharacteristic
363C8—42no—
364C8—122no—
365C2×C3—22no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3#200911yescharacteristic
369C2×C3#200911yescharacteristic
370C2×C3#202211yescharacteristic
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—82no—
380C22#505312yes—
381C22#522311yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—242no—
391C4—241no—
392C4—241no—
393C4—481no—
394C3#998511yescharacteristic, complemented, Sylow
395C2—22no—
396C2—41no—
397C2—41no—
398C2#7170611yescharacteristic
399C2#7211111yescharacteristic
400C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3095C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#3095D3
hypocenterC3#9985
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
394. C3#9985semidirect

Maximal subgroups

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

subgroupindex
C3×#30952
#828082
#822202
#716892
#715942
#699812
#676862
#99853