Finite Group Databasebeta

← #10698359

Subgroups — #10698359

Order 768.

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

normal subgroups only

This group has 422 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#73359C211yesmaximal, characteristic
2#69946C211yesmaximal, characteristic
3#69529C211yesmaximal, characteristic
4C2×#4452C211yesmaximal, characteristic
5C3×#1294C211yesmaximal, characteristic
6#67677C211yesmaximal, characteristic
7#67625C211yesmaximal, characteristic
8#10377—31nomaximal, Sylow
9C2×#924C2211yescharacteristic
10C2×C3×C22⋊C8C2211yescharacteristic
11C3×#380C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13#5212—21no—
14#5212C2211yescharacteristic
15#5210—21no—
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4661—21no—
19#4660—21no—
20#4655—21no—
21#4654—21no—
22#4527—21no—
23#4527—21no—
24#4467—21no—
25#4467—21no—
26#4452—21no—
27#4452—21no—
28#2569—31no—
29#1996—31no—
30#1520—31no—
31C2×#328—31no—
32#1294D311yescharacteristic
33#1289—31no—
34#1282—31no—
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C22×Dic6Q811yescharacteristic
38C22×C3×C8—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×#217—21no—
42C2×#217—21no—
43C2×#217D411yescharacteristic
44C2×#217D411yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3Q811yescharacteristic
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×C4×Dic3—21no—
57C22×C3⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95D411yescharacteristic
63C3×#95D411yescharacteristic
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71#830—21no—
72#830—21no—
73C2×C22⋊Q8—31no—
74C2×C22⋊C8—31no—
75C2×C22⋊C8C2×D311yescharacteristic
76#400—61no—
77#398—61no—
78#387—61no—
79#387—61no—
80#385—61no—
81#380—31no—
82#380—61no—
83#380C2×D311yescharacteristic
84C4×C22⋊C4—31no—
85C2×#95C2×D311yescharacteristic
86#342—61no—
87#342—61no—
88#328—61no—
89#328—61no—
90#328—61no—
91#328—61no—
92C24×C3C2×D411yescharacteristic
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—42no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—81no—
100C22×C3×C4—81no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×Q811yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—81no—
116C2×C3×C8—41no—
117C2×C3×C8—81no—
118C3×C4⋊C4—81no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121#217—21no—
122#217—21no—
123#217—21no—
124#217—21no—
125#217—41no—
126#217—41no—
127#217—41no—
128#217—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—42no—
132C4⋊Dic3—81no—
133C4⋊Dic3—81no—
134Dic3⋊C4—42no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137Dic3⋊C4—81no—
138C4×Dic3—81no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—81no—
141C22×Q8—31no—
142C23×C4—21no—
143C23×C4C22×D311yescharacteristic
144C22×C8—21no—
145C22×C8—61no—
146C22⋊Q8—61no—
147C22⋊Q8—61no—
148C22⋊Q8—61no—
149C22⋊Q8—61no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—31no—
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×C22⋊C4—21no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C42—61no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167C22⋊C8—61no—
168#95—21no—
169#95—21no—
170#95—31no—
171#95—31no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177#95D1211yescharacteristic
178#95D1211yescharacteristic
179C23×C3—41no—
180C23×C3—41no—
181C23×C3#12011yescharacteristic
182C23×C3C22⋊Q811yescharacteristic
183C23×C3#12411yescharacteristic
184C23×C3#13511yescharacteristic
185C23×C3#13511yescharacteristic
186C23×C3#13711yescharacteristic
187C23×C3#13711yescharacteristic
188C2×C3×C4—22no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
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—41no—
200C2×C3×C4—82no—
201C2×C3×C4—82no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4C22⋊Q811yescharacteristic
206C2×C3×C4C22⋊Q811yescharacteristic
207C2×C3×C4#12411yescharacteristic
208C2×C3×C4#12411yescharacteristic
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225Dic6—81no—
226Dic6—81no—
227C3×C8—81no—
228C3⋊C8—81no—
229C24D3×D411yescharacteristic
230C2×Q8—61no—
231C2×Q8—61no—
232C2×Q8—241no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×D1211yescharacteristic
248C22×C4#23511yescharacteristic
249C22×C4#23511yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4#23711yescharacteristic
252C22×C4D3×Q811yescharacteristic
253C2×C8—41no—
254C2×C8—81no—
255C2×C8—121no—
256C2×C8—241no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—81no—
260C4⋊C4—122no—
261C4⋊C4—122no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—241no—
265C4⋊C4—241no—
266C4⋊C4—241no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C42—241no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3#39311yescharacteristic
289C22×C3#45211yescharacteristic
290C22×C3#45211yescharacteristic
291C22×C3#47911yescharacteristic
292C22×C3#47911yescharacteristic
293C22×C3#48811yescharacteristic
294C22×C3#49111yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—42no—
300C3×C4—81no—
301C3×C4—161no—
302Dic3—81no—
303Dic3—81no—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307Dic3—81no—
308C23—41no—
309C23—41no—
310C23#88211yescharacteristic
311C23#88511yescharacteristic
312C23#89611yescharacteristic
313C23#91211yescharacteristic
314C23#91511yescharacteristic
315C23#91611yescharacteristic
316C23#91811yescharacteristic
317Q8—241no—
318Q8—241no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4#87511yescharacteristic
353C2×C4#88511yescharacteristic
354C2×C4#89611yescharacteristic
355C2×C4#89711yescharacteristic
356C8—81no—
357C8—241no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#202611yescharacteristic
365C2×C3#203111yescharacteristic
366C2×C3#203211yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22#467311yescharacteristic
378C22#476811yescharacteristic
379C22#478311yescharacteristic
380C22#479711yescharacteristic
381C22#480811yescharacteristic
382C22#485711yescharacteristic
383C22#487711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—81no—
390C4—161no—
391C4—241no—
392C4—241no—
393C4—241no—
394C4—241no—
395C4—241no—
396C4—241no—
397C3#1037711yescharacteristic, complemented, Sylow
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2#6877911yescharacteristic
405C2#6891111yescharacteristic
406C2#6896711yescharacteristic
subgroupG/H
centerC22#4673
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#1294D3
hypocenterC3#10377
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
397. C3#10377semidirect

Maximal subgroups

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

subgroupindex
#733592
#699462
#695292
C2×#44522
C3×#12942
#676772
#676252
#103773