Finite Group Databasebeta

← #10696592

Subgroups — #10696592

Order 768.

542 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2641C211yesmaximal, characteristic
2#83159C211yesmaximal, characteristic
3#83021C211yesmaximal, characteristic, complemented
4#71760C211yesmaximal, characteristic
5#71735C211yesmaximal, characteristic, complemented
6#69433C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10097—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#394—22no—
13C3×#394—22no—
14C3×#390—21no—
15C3×#390—21no—
16C3×#376C2211yescharacteristic
17#5215—22no—
18#5071—22no—
19#5070—21no—
20#5070—21no—
21#5069—21no—
22#5064—21no—
23#5063—22no—
24#4971—21no—
25#4970—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4544—22no—
30#4543—22no—
31#4452—22no—
32#3236—31no—
33#3138—31no—
34#2641D311yescharacteristic
35#1555—31no—
36#1528—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C3×#127—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—24no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58#926—21no—
59#924—21no—
60#924—21no—
61C2×#208—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#95—21no—
68C3×#95D412yes—
69#832—24no—
70#832—24no—
71#830—21no—
72#830—21no—
73#810—22no—
74#810—22no—
75#808—22no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×#127C2×D311yescharacteristic
85C2×#125C2×D311yescharacteristic
86#501—61no—
87#491—61no—
88#490—62no—
89#486—61no—
90#486—61no—
91#484—61no—
92#483—61no—
93#479—62no—
94#432—31no—
95#432—31no—
96#418—62no—
97C4×M16—31no—
98#394—22no—
99#394—22no—
100#390—21no—
101#390—21no—
102#376C2×D311yescharacteristic
103#331—62no—
104#330—62no—
105#328—62no—
106C24×C3—22no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—44no—
110C2×C3×D4—44no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—82no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×Dic6—21no—
123C2×Dic6—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—84no—
133C3×C22⋊C4—84no—
134C3×C42—42no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147#208—42no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4#6712yes—
154C2×M16—61no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157#127—21no—
158#127—21no—
159#127—21no—
160#127—21no—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—21no—
165#125—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170#117—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4#6711yescharacteristic
174C2×C4⋊C4#6711yescharacteristic
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C42—21no—
178C2×C42C22×D311yescharacteristic
179#107—62no—
180#106—62no—
181C4⋊C8—32no—
182C4⋊C8—61no—
183#103—62no—
184#103—62no—
185#102—64no—
186#102—64no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189#97—32no—
190#97—61no—
191#97—61no—
192C4×C8—61no—
193#95—21no—
194#95#6712yes—
195C23×C3—44no—
196C23×C3—44no—
197C23×C3—42no—
198C23×C3—44no—
199C23×C3#12011yescharacteristic
200C3×D4—44no—
201C3×D4—82no—
202C3×D4—82no—
203C3×D4—82no—
204C3×D4—82no—
205C3×D4—82no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×Dic3—41no—
223C2×Dic3—41no—
224Dic6—42no—
225Dic6—81no—
226C3⋊C8—42no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C24—22no—
230C2×Q8—61no—
231C2×Q8—61no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—82no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4C2×#6711yescharacteristic
245C22×C4C2×#6711yescharacteristic
246C22×C4C2×#6711yescharacteristic
247M16—122no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—121no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C22⋊C4—121no—
268C42—42no—
269C42—121no—
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—88no—
277C22×C3—82no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3#39211yescharacteristic
281C22×C3#45512yes—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—82no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4#45412yes—
291Dic3—81no—
292Dic3—81no—
293C23—44no—
294C23—44no—
295C23—42no—
296C23—44no—
297C23#95311yescharacteristic
298Q8—122no—
299Q8—241no—
300D4—44no—
301D4—82no—
302D4—82no—
303D4—82no—
304D4—82no—
305D4—82no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4#93711yescharacteristic
319C2×C4#93711yescharacteristic
320C2×C4#93811yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#93911yescharacteristic
323C2×C4#94011yescharacteristic
324C8—122no—
325C8—122no—
326C8—241no—
327C2×C3—22no—
328C2×C3—162no—
329C2×C3#200311yescharacteristic
330C2×C3#200411yescharacteristic
331C2×C3#200811yescharacteristic
332C22—21no—
333C22—21no—
334C22—88no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22#522311yescharacteristic
339C22#524212yes—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—82no—
346C4—81no—
347C4—81no—
348C4—241no—
349C4—241no—
350C4#506212yes—
351C3#1009711yescharacteristic, complemented, Sylow
352C2—22no—
353C2—162no—
354C2#7173011yescharacteristic
355C2#7185711yescharacteristic
356C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2641C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#2641D3
hypocenterC3#10097
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2641

Complements

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

NH ≅ G/Ntype
3. #83021C2semidirect
5. #71735C2semidirect
6. #69433C2semidirect
7. #67615C2semidirect
351. C3#10097semidirect

Maximal subgroups

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

subgroupindex
C3×#26412
#831592
#830212
#717602
#717352
#694332
#676152
#100973