Finite Group Databasebeta

← #10697455

Subgroups — #10697455

Order 768.

484 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 95 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#83370C211yesmaximal, characteristic, complemented
2#83335C211yesmaximal, characteristic
3#79269C211yesmaximal, characteristic
4C3×#1518C211yesmaximal, characteristic, complemented
5#69555C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic
7#67741C211yesmaximal, characteristic, complemented
8#10223—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5093—21no—
15#5092—21no—
16#5089—21no—
17#5089—21no—
18#5069—21no—
19#5067—21no—
20#5050—22no—
21#5031—22no—
22#5002—22no—
23#4925C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4842—22no—
26#4735—22no—
27#4694—21no—
28#4680—21no—
29C3×#328—22no—
30#4529—22no—
31#4451—22no—
32#3227—31no—
33#3209—31no—
34#3135—31no—
35#1993—31no—
36#1518D311yescharacteristic, complemented
37#1517—31no—
38#1281—31no—
39C2×#235—21no—
40C22×Dic6—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—24no—
52#930—21no—
53#930—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C22×C3⋊C8D412yes—
57#875—21no—
58#874D411yescharacteristic
59C4×D12—21no—
60C4×D12—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65#869D412yes—
66C3×C4⋊C8—21no—
67C3×C22⋊C8D412yes—
68C3×C4×C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71#830D412yes—
72#821—24no—
73#817—24no—
74#810—22no—
75#810—22no—
76#810—24no—
77#809—22no—
78#809—22no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#532—31no—
85C2×C4⋊Q8—31no—
86#498—61no—
87#494—61no—
88#491—61no—
89#490—61no—
90#489—61no—
91#489—61no—
92#487—61no—
93#486—61no—
94#481—62no—
95#476—62no—
96#462—62no—
97#451—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#416—62no—
101C2×#97—31no—
102#382C2×D311yescharacteristic
103#340—62no—
104#328—22no—
105#327—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113C22×Dic3—41no—
114#235—42no—
115#235—42no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×C4×D3—41no—
120C2×Dic6—21no—
121C2×Dic6—22no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—42no—
132C3×C22⋊C4—41no—
133C3×C42—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#216—84no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—22no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×#41—61no—
152C22×Q8—61no—
153C2×Q16—64no—
154C22×C8—32no—
155C4⋊Q8—32no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8D1211yescharacteristic
161C4⋊Q8D1211yescharacteristic
162#127—31no—
163#125—21no—
164#124—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167#121—61no—
168#121—61no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×D4—61no—
172C4×D4—61no—
173#117—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#106—64no—
179C4⋊C8—21no—
180C4⋊C8—61no—
181#103—62no—
182#103—62no—
183#103—64no—
184#102—62no—
185#102—62no—
186#102—64no—
187C22⋊C8—32no—
188C22⋊C8#6712yes—
189#97—61no—
190#97—61no—
191C4×C8—21no—
192C4×C8—61no—
193#95—21no—
194#95—21no—
195C23×C3#12711yescharacteristic
196C22×D3—41no—
197C3×Q8—42no—
198C3×Q8—81no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#13512yes—
213C2×C3×C4#13712yes—
214#67—81no—
215#67—81no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—82no—
219D12—42no—
220D12—42no—
221D12—81no—
222C4×D3—82no—
223Dic6—44no—
224Dic6—42no—
225Dic6—82no—
226C3×C8—42no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229#41—122no—
230#41—122no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—61no—
234C2×Q8—62no—
235C2×Q8—121no—
236C2×Q8—242no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—121no—
241C22×C4—41no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4C2×#6711yescharacteristic
247Q16—244no—
248C2×C8—22no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8—62no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—62no—
261C4⋊C4—61no—
262C4⋊C4—62no—
263C4⋊C4—61no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C22⋊C4—41no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—41no—
270C42C2×D1211yescharacteristic
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#39211yescharacteristic
277C22×C3#49511yescharacteristic
278C22×C3#49711yescharacteristic
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4—161no—
289C3×C4#45212yes—
290C3×C4#46412yes—
291Dic3—82no—
292Dic3—81no—
293C23—121no—
294C23#94011yescharacteristic
295Q8—42no—
296Q8—81no—
297Q8—124no—
298Q8—122no—
299Q8—242no—
300D4—122no—
301D4—122no—
302D4—241no—
303D4—241no—
304D4—241no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—121no—
313C2×C4—122no—
314C2×C4—242no—
315C2×C4—242no—
316C2×C4#88211yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#88411yescharacteristic
320C2×C4#91612yes—
321C2×C4#91812yes—
322C2×C4#93011yescharacteristic
323C2×C4#93711yescharacteristic
324C8—42no—
325C8—122no—
326C8—122no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3#200811yescharacteristic
330C2×C3#200911yescharacteristic
331C2×C3#201511yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#496011yescharacteristic
339C22#516811yescharacteristic
340C22#519011yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—161no—
348C4—242no—
349C4—241no—
350C4#479812yes—
351C4#480812yes—
352C3#1022311yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—481no—
356C2#7005311yescharacteristic
357C2#7027811yescharacteristic
358C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1518D3
hypocenterC3#10223
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
1. #83370C2semidirect
4. C3×#1518C2semidirect
5. #69555C2semidirect
7. #67741C2semidirect
36. #1518D3semidirect
352. C3#10223semidirect

Maximal subgroups

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

subgroupindex
#833702
#833352
#792692
C3×#15182
#695552
#695452
#677412
#102233