Finite Group Databasebeta

← #10900915

Subgroups — #10900915

Order 768.

1358 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 171 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#85735C211yesmaximal, characteristic
2C3×#2484C211yesmaximal, characteristic, complemented
3#82192C212yesmaximal, complemented
4#71594C211yesmaximal, characteristic, complemented
5#69596C214yesmaximal, complemented
6#69539C214yesmaximal, complemented
7#69537C212yesmaximal, complemented
8#29637—31nomaximal, Sylow
9C22×#235C2212yes—
10C2×C3×#117C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#929C2214yes—
13#5526—22no—
14#5525—22no—
15#5524—24no—
16#5510—21no—
17#5510—21no—
18#5493C2214yes—
19#5477—22no—
20C3×#400—21no—
21C3×#398—21no—
22C3×#387—22no—
23C3×#385C2212yes—
24C3×#381C2211yescharacteristic
25C3×#381C2212yes—
26#5213C2214yes—
27#5211C2212yes—
28#4991—22no—
29#4989—24no—
30#4979—22no—
31C2×#806C2218yes—
32#4526C2214yes—
33#3449—31no—
34#2484D311yescharacteristic, complemented
35#2406—32no—
36#2044—34no—
37#2022—34no—
38#1820—31no—
39#1782—32no—
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67C2314yes—
44C2×#235—22no—
45C2×#235—22no—
46C2×#235—22no—
47C2×#235—22no—
48C2×#235—48no—
49C22×D12D412yes—
50C22×C4×D3D414yes—
51C22×Dic6D412yes—
52C3×#117—22no—
53C3×#117—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4C2312yes—
62C2×C3×C22⋊C4C2312yes—
63C2×C3×C42D412yes—
64C2×#217C2314yes—
65#930—24no—
66#930—24no—
67#929—48no—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—24no—
70C2×Dic3⋊C4—24no—
71#897—24no—
72#897—24no—
73#895—24no—
74#895—24no—
75#886—28no—
76#882—28no—
77#875—24no—
78#874—22no—
79C4×D12—24no—
80C4×D12—24no—
81#869—22no—
82C3×#95—21no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—21no—
86C3×#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—24no—
90#806—48no—
91C22×#41—32no—
92#537—61no—
93#537—61no—
94#536—62no—
95#535—62no—
96#534—34no—
97#532—62no—
98#525—64no—
99C2×#123—34no—
100C2×#117C2×D311yescharacteristic
101C22×C4⋊C4C2×D311yescharacteristic
102C2×#99—38no—
103#400—21no—
104#398—21no—
105#395—62no—
106#394—62no—
107#387—22no—
108#386—34no—
109#385—64no—
110#385C2×D312yes—
111#381—32no—
112#381C2×D312yes—
113#381C2×D311yescharacteristic
114#342—34no—
115C24×C3C2411yescharacteristic
116C23×D3C2×D414yes—
117C22×C3×C4—24no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—24no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C2×#67—24no—
132C2×#67—24no—
133C2×#67—28no—
134C2×#67—28no—
135C22×Dic3—24no—
136C22×Dic3C2×D414yes—
137#235—48no—
138#235—48no—
139#235—48no—
140#235—48no—
141C2×D12—22no—
142C2×D12—22no—
143C2×D12—24no—
144C2×D12—24no—
145C2×D12—44no—
146C2×C4×D3—24no—
147C2×C4×D3—24no—
148C2×C4×D3—44no—
149C2×C4×D3—44no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—44no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—44no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—42no—
159C3×C22⋊C4—24no—
160C3×C22⋊C4—24no—
161C3×C42—22no—
162C3×C42—22no—
163#217—28no—
164(C2×D3)⋊C4—28no—
165(C2×D3)⋊C4—28no—
166(C2×D3)⋊C4—48no—
167(C2×D3)⋊C4—48no—
168C4⋊Dic3—24no—
169C4⋊Dic3—24no—
170Dic3⋊C4—48no—
171C2×#41—62no—
172C2×#41—62no—
173C2×#41—62no—
174C2×#41—62no—
175C2×#41—128no—
176C22×Q8—32no—
177C22×D4—34no—
178C22×D4—32no—
179C23×C4—34no—
180C23×C4C22×D311yescharacteristic
181C23×C4C22×D311yescharacteristic
182C23×C4C22×D311yescharacteristic
183C4⋊Q8—62no—
184#127—62no—
185#124—64no—
186#123—68no—
187#123—128no—
188C22⋊Q8—64no—
189C22⋊Q8—64no—
190#121—64no—
191#121—64no—
192#121—64no—
193#121—64no—
194#120—68no—
195C4×D4—64no—
196C4×D4—64no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—64no—
205C2×C4⋊C4#6711yescharacteristic
206C2×C4⋊C4#6711yescharacteristic
207C2×C22⋊C4—34no—
208C2×C22⋊C4—64no—
209C2×C22⋊C4—64no—
210C2×C22⋊C4C22×D312yes—
211C2×C22⋊C4C22×D312yes—
212C2×C42#6712yes—
213#99—128no—
214#95—21no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—64no—
220#95—62no—
221#95—62no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3C22×D412yes—
225C23×C3C22×D411yescharacteristic
226C23×C3H(5,𝔽2)14yes—
227C22×D3—24no—
228C22×D3—24no—
229C22×D3—24no—
230C22×D3—44no—
231C22×D3—44no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—44no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12012yes—
254#67—48no—
255#67—48no—
256C2×Dic3—24no—
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—44no—
260C2×Dic3—44no—
261C2×Dic3—44no—
262D12—44no—
263D12—44no—
264D12—84no—
265D12—84no—
266C4×D3—48no—
267C4×D3—48no—
268Dic6—44no—
269Dic6—44no—
270C24—34no—
271C24C23×D311yescharacteristic
272#41—128no—
273#41—128no—
274#41—128no—
275#41—128no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—124no—
279C2×D4—64no—
280C2×D4—64no—
281C2×D4—64no—
282C2×D4—68no—
283C2×D4—68no—
284C2×D4—64no—
285C2×D4—62no—
286C2×D4—62no—
287C2×D4—124no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—34no—
296C22×C4—42no—
297C22×C4—64no—
298C22×C4—64no—
299C22×C4—64no—
300C22×C4—124no—
301C22×C4—124no—
302C22×C4D3×D412yes—
303C22×C4D3×D412yes—
304C22×C4C2×#6711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306C22×C4C2×#6712yes—
307C22×C4C2×#6712yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—64no—
315C4⋊C4—64no—
316C4⋊C4—128no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—68no—
320C22⋊C4—68no—
321C22⋊C4—68no—
322C22⋊C4—128no—
323C22⋊C4—128no—
324C42—22no—
325C42—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—44no—
331C22×C3C2×#12011yescharacteristic
332C22×C3#53414yes—
333C22×C3#53512yes—
334C2×D3—48no—
335C2×D3—44no—
336C2×D3—44no—
337C2×D3—44no—
338C2×D3—84no—
339C3×C4—21no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—42no—
343C3×C4—42no—
344C3×C4—42no—
345C3×C4—82no—
346C3×C4—81no—
347Dic3—44no—
348Dic3—84no—
349C23—41no—
350C23—41no—
351C23—64no—
352C23—64no—
353C23—64no—
354C23—124no—
355C23—124no—
356C23C2×D3×D412yes—
357C23#100914yes—
358C23C22×#6711yescharacteristic
359Q8—124no—
360Q8—124no—
361D4—124no—
362D4—124no—
363D4—128no—
364D4—128no—
365D4—244no—
366D4—244no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—64no—
387C2×C4—64no—
388C2×C4—64no—
389C2×C4—128no—
390C2×C4—128no—
391C2×C4—124no—
392C2×C4—124no—
393C2×C4—124no—
394C2×C4#93711yescharacteristic
395C2×C4#93711yescharacteristic
396C2×C4#93712yes—
397C2×C3—24no—
398C2×C3—22no—
399C2×C3#261311yescharacteristic
400C2×C3#302511yescharacteristic
401C2×C3#302811yescharacteristic
402D3—84no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—22no—
407C22—44no—
408C22—128no—
409C22—124no—
410C22—124no—
411C22—124no—
412C22—244no—
413C22#565314yes—
414C22C2×#93711yescharacteristic
415C22#583012yes—
416C4—21no—
417C4—22no—
418C4—21no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—82no—
423C4—81no—
424C4—124no—
425C4—244no—
426C3#2963711yescharacteristic, complemented, Sylow
427C2—22no—
428C2—24no—
429C2—244no—
430C2#8290311yescharacteristic
431C2#8293611yescharacteristic
432C2#8341711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2484C2
socleC22×C3C2×#120
prime radical#85735C2
hypercenter#2484D3
hypocenterC3#29637
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
2. C3×#2484C2semidirect
3. #82192C2semidirect
4. #71594C2semidirect
5. #69596C2semidirect
6. #69539C2semidirect
7. #69537C2semidirect
34. #2484D3semidirect
426. C3#29637semidirect

Maximal subgroups

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

subgroupindex
#857352
C3×#24842
#821922
#715942
#695962
#695392
#695372
#296373