Finite Group Databasebeta

← #10696476

Subgroups — #10696476

Order 768.

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

normal subgroups only

This group has 552 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
1C2×C3×#490C211yesmaximal, characteristic
2C2×#5071C211yesmaximal, characteristic
3#83055C211yesmaximal, characteristic, complemented
4#71765C211yesmaximal, characteristic
5C2×#4543C211yesmaximal, characteristic, complemented
6C2×#4465C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10080—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#490—22no—
13C3×#490—22no—
14C2×C3×#102—21no—
15C2×C3×#102—21no—
16C2×C3×#97C2211yescharacteristic
17C2×#832—21no—
18C2×#832—21no—
19C2×#832—22no—
20#5071—22no—
21#5071—22no—
22#4986—21no—
23#4985—21no—
24#4983—21no—
25#4982—22no—
26#4981—21no—
27#4936C2211yescharacteristic
28C2×#803C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4543—22no—
31#4465—22no—
32C2×#490—31no—
33C2×#490D311yescharacteristic
34#2822—31no—
35#1961—31no—
36C2×#330—31no—
37C2×#330—31no—
38#1295—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×C8—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127D412yes—
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C3×#125—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D412yes—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C3×#102—24no—
69C3×#102—24no—
70C3×#97—21no—
71C3×#97D412yes—
72#832—24no—
73#832—24no—
74#832—44no—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#803—21no—
81#803—21no—
82#803—21no—
83#803D412yes—
84C2×#127C2×D311yescharacteristic
85C2×#125—31no—
86C2×#125C2×D311yescharacteristic
87#490—22no—
88#490—22no—
89#490—62no—
90#490—62no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—61no—
94C2×#102—61no—
95C2×#102—62no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99#400—61no—
100#400—61no—
101#398—61no—
102#398—62no—
103#398—61no—
104#382—31no—
105#330—62no—
106#330—62no—
107C24×C3—41no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—42no—
112C2×C3×D4—44no—
113C2×C3×D4—42no—
114C2×C3×D4—44no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—22no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C2×C3×C8—22no—
127C2×C3×C8—42no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—42no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—82no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—42no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×C8—21no—
159C22×C8—61no—
160C22×C8—61no—
161#127—21no—
162#127—21no—
163#127—21no—
164#127#6712yes—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—21no—
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4#6711yescharacteristic
182C2×C4⋊C4#6711yescharacteristic
183C2×C42C22×D311yescharacteristic
184#102—24no—
185#102—24no—
186#102—64no—
187#102—64no—
188#102—124no—
189#97—21no—
190#97—32no—
191#97—61no—
192#97—61no—
193#97—61no—
194#97#6712yes—
195#95—62no—
196#95—62no—
197#95—62no—
198#95—61no—
199#95—61no—
200C23×C3—42no—
201C23×C3—81no—
202C23×C3—82no—
203C23×C3—81no—
204C23×C3—82no—
205C23×C3#12011yescharacteristic
206C3×D4—44no—
207C3×D4—44no—
208C3×D4—44no—
209C3×D4—82no—
210C3×D4—82no—
211C3×D4—82no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C3×C8—42no—
236C3⋊C8—42no—
237C3⋊C8—42no—
238C24—41no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4C2×#6711yescharacteristic
257C2×C8—22no—
258C2×C8—42no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—122no—
262C2×C8—121no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—122no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—242no—
282C42D3×D411yescharacteristic
283C42#23711yescharacteristic
284C42C2×#6711yescharacteristic
285C42C2×#6711yescharacteristic
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—162no—
290C22×C3—162no—
291C22×C3#39211yescharacteristic
292C22×C3#45311yescharacteristic
293C22×C3#45412yes—
294C22×C3#45411yescharacteristic
295C22×C3#45412yes—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—81no—
303C3×C4—81no—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307C23—42no—
308C23—81no—
309C23—82no—
310C23—81no—
311C23—82no—
312C23#93711yescharacteristic
313D4—44no—
314D4—44no—
315D4—44no—
316D4—82no—
317D4—82no—
318D4—82no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4#93711yescharacteristic
337C2×C4#93811yescharacteristic
338C2×C4#93911yescharacteristic
339C2×C4#93911yescharacteristic
340C2×C4#94011yescharacteristic
341C2×C4#95311yescharacteristic
342C8—42no—
343C8—122no—
344C8—122no—
345C2×C3—162no—
346C2×C3#165712yes—
347C2×C3#165812yes—
348C2×C3#200311yescharacteristic
349C2×C3#200311yescharacteristic
350C2×C3#200611yescharacteristic
351C22—82no—
352C22—82no—
353C22—82no—
354C22—162no—
355C22—162no—
356C22#506212yes—
357C22#519811yescharacteristic
358C22#520112yes—
359C22#520111yescharacteristic
360C22#522311yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—81no—
369C4—241no—
370C4—242no—
371C4—241no—
372C3#1008011yescharacteristic, complemented, Sylow
373C2—162no—
374C2#7124912yes—
375C2#7125012yes—
376C2#7173011yescharacteristic
377C2#7209611yescharacteristic
378C2#7209811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#490C2
socleC23×C3#120
prime radicalC2×C3×#127C22
hypercenterC2×#490D3
hypocenterC3#10080
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
3. #83055C2semidirect
5. C2×#4543C2semidirect
6. C2×#4465C2semidirect
7. #67659C2semidirect
372. C3#10080semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4902
C2×#50712
#830552
#717652
C2×#45432
C2×#44652
#676592
#100803