Finite Group Databasebeta

← #10696054

Subgroups — #10696054

Order 768.

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

normal subgroups only

This group has 560 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
1#79310C211yesmaximal, characteristic, complemented
2#79298C211yesmaximal, characteristic
3#73419C211yesmaximal, characteristic
4#70770C211yesmaximal, characteristic, complemented
5#70762C211yesmaximal, characteristic
6#69441C211yesmaximal, characteristic, complemented
7C3×#1274C211yesmaximal, characteristic, complemented
8#10016—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#376C2211yescharacteristic
15C4×(C2×D3)⋊C4—22no—
16#4924C2211yescharacteristic
17#4850—21no—
18#4849—21no—
19#4842—21no—
20#4841—21no—
21#4739—21no—
22#4731—21no—
23#4722—21no—
24#4721—21no—
25C4×Dic12—22no—
26C4×#204—22no—
27#4674—21no—
28#4673—21no—
29#4672—21no—
30#4659—21no—
31#4523—22no—
32#4459—22no—
33#4456—22no—
34#3104—31no—
35#3101—31no—
36#2398—31no—
37#1995—31no—
38#1487—31no—
39#1486—31no—
40#1274D311yescharacteristic, complemented
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C22×Dic6D411yescharacteristic
44C2×C3×C42D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—21no—
49#926—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52#924—21no—
53#924—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C2×#208—21no—
57C2×Dic12—22no—
58C2×#204—22no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8D412yes—
77C3×C4×C8D412yes—
78C3×#95—21no—
79C3×#95—21no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82#821—22no—
83#818—22no—
84#817—22no—
85#816—22no—
86#816—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90#533—31no—
91C2×C4×Q8—31no—
92C2×C4×D4—31no—
93#484—61no—
94#483—61no—
95#477—61no—
96#476—61no—
97#462—61no—
98#461—61no—
99#451—61no—
100#450—61no—
101C4×Q16—62no—
102C4×SD16—62no—
103#431C2×D311yescharacteristic
104#431C2×D311yescharacteristic
105#423—31no—
106#400—61no—
107#397—61no—
108#394—61no—
109#393—61no—
110C4×C22⋊C4—62no—
111#376C2×D311yescharacteristic
112#339—62no—
113Q8⋊C8—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C22×C3×C4—22no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—42no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—42no—
129C2×C4×D3—42no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—42no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C3×C42—22no—
138C3×C42—22no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—41no—
144(C2×D3)⋊C4—44no—
145(C2×D3)⋊C4—44no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163#208—42no—
164C2×C3⋊C8—41no—
165Dic12—44no—
166#204—44no—
167C22×Q8—31no—
168C22×D4—31no—
169C23×C4—61no—
170C2×Q16—62no—
171C2×SD16—62no—
172C2×M16—61no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175#125—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188#117—61no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—61no—
195C2×C42—61no—
196C2×C42D1212yes—
197C2×C42C22×D311yescharacteristic
198#107—62no—
199#106—62no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—62no—
205#103—62no—
206#103—62no—
207#102—62no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C4×C8D1212yes—
211C4×C8D1212yes—
212#95—21no—
213#95—21no—
214#95—62no—
215#95—62no—
216C23×C3#12011yescharacteristic
217C22×D3—22no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4#13512yes—
234C2×C3×C4#13712yes—
235C2×Dic3—22no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244D12—22no—
245D12—22no—
246D12—41no—
247D12—41no—
248C4×D3—82no—
249C4×D3—81no—
250C4×D3—81no—
251Dic6—22no—
252Dic6—22no—
253Dic6—41no—
254Dic6—41no—
255Dic6—42no—
256Dic6—42no—
257C3×C8—42no—
258C3×C8—42no—
259C3⋊C8—81no—
260C24—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—122no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—122no—
269C22×C4—22no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—122no—
276C22×C4—122no—
277C22×C4C2×D1211yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280Q16—124no—
281SD16—124no—
282M16—122no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—121no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—124no—
299C22⋊C4—124no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C42—22no—
304C42—22no—
305C42—121no—
306C42—121no—
307C42—121no—
308C42C2×D1211yescharacteristic
309C42C2×D1211yescharacteristic
310C42D3×D411yescharacteristic
311C42#23911yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3#39211yescharacteristic
315C22×C3#45212yes—
316C2×D3—44no—
317C2×D3—42no—
318C2×D3—82no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—82no—
326C3×C4#46412yes—
327C3×C4#49511yescharacteristic
328C3×C4#49711yescharacteristic
329Dic3—42no—
330Dic3—81no—
331Dic3—81no—
332Dic3—81no—
333Dic3—81no—
334C23—62no—
335C23—122no—
336C23—121no—
337C23—121no—
338C23—122no—
339C23#88211yescharacteristic
340Q8—62no—
341Q8—62no—
342Q8—121no—
343Q8—121no—
344Q8—122no—
345Q8—122no—
346D4—62no—
347D4—62no—
348D4—121no—
349D4—121no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—62no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—122no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4—242no—
367C2×C4#87411yescharacteristic
368C2×C4#88211yescharacteristic
369C2×C4#88211yescharacteristic
370C2×C4#89511yescharacteristic
371C2×C4#89511yescharacteristic
372C2×C4#89511yescharacteristic
373C2×C4#90412yes—
374C2×C4#90912yes—
375C8—42no—
376C8—42no—
377C8—241no—
378C2×C3—22no—
379C2×C3#201011yescharacteristic
380C2×C3#201011yescharacteristic
381C2×C3#201511yescharacteristic
382D3—82no—
383C22—21no—
384C22—21no—
385C22—124no—
386C22—122no—
387C22—242no—
388C22—241no—
389C22—241no—
390C22#467111yescharacteristic
391C22#473312yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—82no—
397C4—122no—
398C4—241no—
399C4—241no—
400C4—241no—
401C4—241no—
402C4#469611yescharacteristic
403C4#471511yescharacteristic
404C4#484512yes—
405C3#1001611yescharacteristic, complemented, Sylow
406C2—22no—
407C2—242no—
408C2#6867711yescharacteristic
409C2#6870311yescharacteristic
410C2#6870311yescharacteristic
subgroupG/H
centerC2×C4#874
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1274D3
hypocenterC3#10016
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
1. #79310C2semidirect
4. #70770C2semidirect
6. #69441C2semidirect
7. C3×#1274C2semidirect
40. #1274D3semidirect
405. C3#10016semidirect

Maximal subgroups

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

subgroupindex
#793102
#792982
#734192
#707702
#707622
#694412
C3×#12742
#100163