Finite Group Databasebeta

← #10698409

Subgroups — #10698409

Order 768.

396 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 77 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×#2667C211yesmaximal, characteristic
2#69533C211yesmaximal, characteristic
3#69529C211yesmaximal, characteristic
4#69445C211yesmaximal, characteristic
5#67628C211yesmaximal, characteristic
6#67626C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic
8#10385—31nomaximal, Sylow
9C2×#924C2211yescharacteristic
10C3×#399—21no—
11C3×#398—21no—
12C3×#398—21no—
13C3×#396—21no—
14C3×#389—21no—
15C3×#387—21no—
16C3×#381C2211yescharacteristic
17C3×#380C2211yescharacteristic
18C3×C4×C22⋊C4C2211yescharacteristic
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#4527—21no—
23#4527—21no—
24#4452—21no—
25#4452—21no—
26#4452—21no—
27#4452—21no—
28#2667D311yescharacteristic
29#1996—31no—
30#1511—31no—
31#1508—31no—
32#1305—31no—
33#1297—31no—
34#1284—31no—
35C23×C3×C4C2311yescharacteristic
36C22×Dic6Q811yescharacteristic
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×C4⋊Dic3Q811yescharacteristic
52#924—21no—
53#924—21no—
54#924—21no—
55#924—21no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C2×#208—21no—
59C2×#208—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95D411yescharacteristic
65C3×#95D411yescharacteristic
66C3×#95D411yescharacteristic
67C3×#95D411yescharacteristic
68C3⋊#95—21no—
69#830—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73C2×C22⋊Q8—31no—
74#407—31no—
75#407—31no—
76#399—21no—
77#398—21no—
78#398—21no—
79#396—21no—
80#389—21no—
81#387—21no—
82#381C2×D311yescharacteristic
83#380C2×D311yescharacteristic
84C4×C22⋊C4—31no—
85C4×C22⋊C4C2×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—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—81no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×Q811yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—81no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—81no—
115C3×C4⋊C4—81no—
116C3×C4⋊C4—81no—
117C3×C4⋊C4—81no—
118C3×C4⋊C4—81no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—81no—
123C3×C42—81no—
124#217—22no—
125#217—41no—
126#217—41no—
127#217—41no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—81no—
133#208—81no—
134#208—81no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C22×Q8—31no—
138C23×C4C22×D311yescharacteristic
139C2×M16—61no—
140C2×M16—61no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—61no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4#6711yescharacteristic
159C2×C22⋊C4#6711yescharacteristic
160C2×C42—21no—
161C2×C42—61no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164C22⋊C8—61no—
165C22⋊C8—61no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—61no—
171#95#6711yescharacteristic
172#95#6711yescharacteristic
173#95#6711yescharacteristic
174#95#6711yescharacteristic
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3#12011yescharacteristic
180C23×C3C22⋊Q811yescharacteristic
181C23×C3#12411yescharacteristic
182C2×C3×C4—22no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
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—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4C22⋊Q811yescharacteristic
207C2×C3×C4C22⋊Q811yescharacteristic
208C2×C3×C4#12411yescharacteristic
209C2×C3×C4#12411yescharacteristic
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—81no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217Dic6—81no—
218Dic6—81no—
219C3⋊C8—81no—
220C3⋊C8—81no—
221C24C2×#6711yescharacteristic
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—241no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—81no—
234C22×C4—121no—
235C22×C4D3×Q811yescharacteristic
236C22×C4#23911yescharacteristic
237C22×C4#23911yescharacteristic
238C22×C4#23911yescharacteristic
239C22×C4C2×#6711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241M16—241no—
242M16—241no—
243C2×C8—121no—
244C2×C8—121no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C22⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—62no—
260C22⋊C4—81no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C42—81no—
265C42—241no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3#39311yescharacteristic
277C22×C3#45511yescharacteristic
278C22×C3#45611yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—42no—
284C3×C4—81no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4—81no—
289Dic3—82no—
290Dic3—81no—
291Dic3—81no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23#94611yescharacteristic
297C23#94711yescharacteristic
298C23#95311yescharacteristic
299Q8—241no—
300Q8—241no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—122no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4#94611yescharacteristic
333C2×C4#94611yescharacteristic
334C2×C4#94711yescharacteristic
335C2×C4#94711yescharacteristic
336C8—241no—
337C8—241no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3#202811yescharacteristic
343C2×C3#203111yescharacteristic
344C2×C3#203211yescharacteristic
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22#523111yescharacteristic
356C22#524211yescharacteristic
357C22#524611yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—242no—
369C4—241no—
370C4—241no—
371C3#1038511yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2—41no—
375C2—41no—
376C2#7184211yescharacteristic
377C2#7189611yescharacteristic
378C2#7190511yescharacteristic
subgroupG/H
centerC22#5231
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2667C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#2667D3
hypocenterC3#10385
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2667

Complements

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

NH ≅ G/Ntype
371. C3#10385semidirect

Maximal subgroups

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

subgroupindex
C3×#26672
#695332
#695292
#694452
#676282
#676262
#676182
#103853