Finite Group Databasebeta

← #10696498

Subgroups — #10696498

Order 768.

526 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2820C211yesmaximal, characteristic
2C2×#5069C211yesmaximal, characteristic, complemented
3C2×#5055C211yesmaximal, characteristic
4C2×#4543C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic
6#69555C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#10083—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#399—21no—
13C3×#397—21no—
14C3×#396—22no—
15C3×#395—21no—
16C3×#394—21no—
17C3×#382C2211yescharacteristic
18C2×#835—21no—
19C2×#832—21no—
20#5069—22no—
21#5069—22no—
22#5055—22no—
23#5055—22no—
24C2×#810—21no—
25C2×#810—21no—
26C2×#803C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4543—22no—
30#4540—22no—
31#4529—22no—
32C2×#491—31no—
33C2×#489—31no—
34#2820D311yescharacteristic
35#1993—31no—
36C2×#330—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42C2311yescharacteristic
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×#95—22no—
71C3×#95—21no—
72C3×#95—21no—
73#835—24no—
74#832—24no—
75#810—24no—
76#810—24no—
77#803—21no—
78#803—21no—
79#803—21no—
80#803—21no—
81#803D412yes—
82#803D412yes—
83C2×C4⋊Q8—31no—
84C2×#125C2×D311yescharacteristic
85C2×#124C2×D311yescharacteristic
86#491—62no—
87#491—62no—
88#489—62no—
89#489—62no—
90C2×#103—61no—
91C2×#103—61no—
92C2×#103—61no—
93C2×#102—61no—
94C2×#97—31no—
95C2×#97—31no—
96C2×#97—31no—
97#399—21no—
98#397—21no—
99#396—22no—
100#395—21no—
101#394—21no—
102#382C2×D311yescharacteristic
103#340—62no—
104#330—62no—
105#329—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—41no—
121C2×Dic6—22no—
122C2×Dic6—44no—
123C2×Dic6—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—82no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—82no—
139C3×C22⋊C4—82no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—42no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—42no—
154C2×C3⋊C8—41no—
155C22×Q8—61no—
156C22×Q8#6711yescharacteristic
157C22×D4#6711yescharacteristic
158C22×C8—61no—
159C22×C8—61no—
160C22×C8—61no—
161C4⋊Q8—32no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—21no—
169#124—21no—
170#124—21no—
171#124—21no—
172#124—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4#6711yescharacteristic
179C2×C4⋊C4#6711yescharacteristic
180C2×C4⋊C4#6712yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C42C22×D311yescharacteristic
186#103—64no—
187#103—64no—
188#103—64no—
189#102—64no—
190#97—32no—
191#97—32no—
192#97—61no—
193#97—61no—
194#97—61no—
195#97—61no—
196#95—22no—
197#95—21no—
198#95—21no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3—41no—
204C23×C3#12011yescharacteristic
205C3×Q8—42no—
206C3×Q8—81no—
207C3×D4—42no—
208C3×D4—81no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229C2×Dic3—42no—
230C2×Dic3—82no—
231Dic6—44no—
232Dic6—82no—
233C3⋊C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C24—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—42no—
240C2×Q8—62no—
241C2×Q8—124no—
242C2×Q8—121no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—42no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—121no—
251C22×C4C2×#6711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254C2×C8—62no—
255C2×C8—62no—
256C2×C8—62no—
257C2×C8—121no—
258C2×C8—122no—
259C2×C8—122no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—82no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42#23711yescharacteristic
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3#39211yescharacteristic
291C22×C3#45412yes—
292C22×C3#45511yescharacteristic
293C22×C3#45512yes—
294C22×C3#45611yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—81no—
303C3×C4—81no—
304C3×C4—81no—
305Dic3—82no—
306C23—41no—
307C23—41no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23#95311yescharacteristic
312Q8—42no—
313Q8—81no—
314Q8—124no—
315Q8—242no—
316D4—42no—
317D4—81no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—122no—
333C2×C4—242no—
334C2×C4#93711yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93811yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#93911yescharacteristic
339C2×C4#94011yescharacteristic
340C8—122no—
341C8—122no—
342C8—122no—
343C2×C3—161no—
344C2×C3#166212yes—
345C2×C3#166412yes—
346C2×C3#200311yescharacteristic
347C2×C3#200411yescharacteristic
348C2×C3#200711yescharacteristic
349C22—82no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22#522311yescharacteristic
355C22#524211yescharacteristic
356C22#524212yes—
357C22#524512yes—
358C22#524611yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—242no—
370C3#1008311yescharacteristic, complemented, Sylow
371C2—161no—
372C2#7185711yescharacteristic
373C2#7186711yescharacteristic
374C2#7188111yescharacteristic
375C2#7195412yes—
376C2#7195712yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2820C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenter#2820D3
hypocenterC3#10083
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2820

Complements

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

NH ≅ G/Ntype
2. C2×#5069C2semidirect
4. C2×#4543C2semidirect
6. #69555C2semidirect
7. #67630C2semidirect
370. C3#10083semidirect

Maximal subgroups

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

subgroupindex
C3×#28202
C2×#50692
C2×#50552
C2×#45432
C2×#45402
#695552
#676302
#100833