Finite Group Databasebeta

← #10696559

Subgroups — #10696559

Order 768.

534 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 83 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#82978C211yesmaximal, characteristic, complemented
2#82947C211yesmaximal, characteristic
3#78528C211yesmaximal, characteristic, complemented
4C3×#1527C211yesmaximal, characteristic, complemented
5#70832C211yesmaximal, characteristic, complemented
6#70823C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10092—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5229—21no—
15#5227—21no—
16#5221—21no—
17#5217—21no—
18#5133—22no—
19#5058—21no—
20#5057—22no—
21#5055—21no—
22#4953—22no—
23#4947—22no—
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26#4714—22no—
27#4694—21no—
28#4681—21no—
29C3×#329—22no—
30#4482—22no—
31#4470—22no—
32#3213—31no—
33#3141—31no—
34#2640—31no—
35#1556—31no—
36#1527D311yescharacteristic, complemented
37#1505—31no—
38#1300—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C2×#235—21no—
42C22×Dic6—21no—
43C2×C3×M16—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—24no—
52C2×#215—22no—
53C2×#213—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D412yes—
58C2×C4×Dic3—21no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D411yescharacteristic
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3×C4×C8—21no—
73#835—22no—
74#832—22no—
75C3⋊#95—24no—
76C3⋊#95—21no—
77C3⋊#95D412yes—
78#830—21no—
79#821—24no—
80#816—24no—
81C4⋊(C3⋊C8)D412yes—
82#803—21no—
83#532—31no—
84C2×C4⋊Q8—31no—
85C2×#124C2×D311yescharacteristic
86#496—61no—
87#489—61no—
88#489—62no—
89#487—61no—
90#486—61no—
91#463—62no—
92#448—61no—
93#447—61no—
94#432—31no—
95#432C2×D311yescharacteristic
96#417—62no—
97C4×M16C2×D311yescharacteristic
98#397—62no—
99#393—62no—
100#391—61no—
101#390—61no—
102#376—31no—
103#332—62no—
104#329—22no—
105#327—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C22×C3×C4—21no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118C22×Dic3—22no—
119C22×Dic3—22no—
120#235—44no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×C4×D3—41no—
125C2×Dic6—21no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—82no—
129C3×M16—42no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—84no—
141#215—82no—
142#213—82no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4×Dic3—42no—
152C2×C3⋊C8—22no—
153C2×#41—61no—
154C22×Q8—61no—
155C22×Q8D1211yescharacteristic
156C22×D4D1211yescharacteristic
157C2×SD16—62no—
158C2×D8—62no—
159C2×M16—21no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#127—31no—
166#124—21no—
167#124—21no—
168#124—21no—
169#124—21no—
170#124—61no—
171#121—61no—
172#121—61no—
173C4×D4—61no—
174C4×D4—61no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—61no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—64no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182C4⋊C8—21no—
183C4⋊C8—32no—
184#103—64no—
185#103—62no—
186#102—64no—
187#102—62no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190#97—21no—
191#97—61no—
192#97#6712yes—
193C4×C8—21no—
194#95—32no—
195#95—64no—
196#95—61no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3#12011yescharacteristic
202C22×D3—41no—
203C3×Q8—42no—
204C3×Q8—81no—
205C3×D4—42no—
206C3×D4—81no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218#67—82no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—44no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225D12—42no—
226D12—42no—
227D12—81no—
228C4×D3—82no—
229Dic6—42no—
230Dic6—82no—
231Dic6—82no—
232C3×C8—42no—
233C3×C8—81no—
234C3⋊C8—42no—
235C24—21no—
236#41—124no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—42no—
240C2×Q8—61no—
241C2×Q8—121no—
242C2×Q8—121no—
243C2×Q8—242no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—42no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—121no—
251C22×C4—21no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—121no—
255C22×C4C2×D1211yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258SD16—242no—
259D8—242no—
260M16—42no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—62no—
264C4⋊C4—61no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—244no—
277C42—122no—
278C42#23511yescharacteristic
279C42D3×D411yescharacteristic
280C42D3×D411yescharacteristic
281C42C2×#6711yescharacteristic
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—84no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#39211yescharacteristic
289C22×C3#45412yes—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—81no—
299C3×C4#45512yes—
300Dic3—82no—
301Dic3—81no—
302Dic3—82no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—42no—
307C23—121no—
308C23#88211yescharacteristic
309Q8—42no—
310Q8—81no—
311Q8—122no—
312Q8—242no—
313Q8—242no—
314D4—42no—
315D4—81no—
316D4—122no—
317D4—122no—
318D4—241no—
319D4—242no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—124no—
328C2×C4—121no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—242no—
332C2×C4#88211yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4#88411yescharacteristic
335C2×C4#93011yescharacteristic
336C2×C4#93711yescharacteristic
337C2×C4#94011yescharacteristic
338C8—42no—
339C8—81no—
340C8—122no—
341C2×C3—22no—
342C2×C3—161no—
343C2×C3#200311yescharacteristic
344C2×C3#200411yescharacteristic
345C2×C3#200811yescharacteristic
346D3—161no—
347C22—21no—
348C22—21no—
349C22—84no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—241no—
354C22—241no—
355C22—241no—
356C22#482712yes—
357C22#496011yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—242no—
365C4—242no—
366C4—241no—
367C4#474612yes—
368C3#1009211yescharacteristic, complemented, Sylow
369C2—22no—
370C2—161no—
371C2—481no—
372C2#7008411yescharacteristic
373C2#7038811yescharacteristic
374C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#392
prime radical#82947C2
hypercenter#1527D3
hypocenterC3#10092
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
1. #82978C2semidirect
3. #78528C2semidirect
4. C3×#1527C2semidirect
5. #70832C2semidirect
6. #70823C2semidirect
7. #67665C2semidirect
36. #1527D3semidirect
368. C3#10092semidirect

Maximal subgroups

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

subgroupindex
#829782
#829472
#785282
C3×#15272
#708322
#708232
#676652
#100923