Finite Group Databasebeta

← #10691130

Subgroups — #10691130

Order 768.

544 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 113 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#79067C211yesmaximal, characteristic
2#74157C211yesmaximal, characteristic, complemented
3#70789C211yesmaximal, characteristic
4#70766C211yesmaximal, characteristic, complemented
5#69545C211yesmaximal, characteristic
6C3×#1286C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9180—31nomaximal, Sylow
9C2×#909—21no—
10C2×#887—21no—
11#5477C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5133—21no—
17#4972—21no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4849C412yescomplemented
21#4848C412yescomplemented
22#4845—22no—
23#4650—21no—
24#4649—21no—
25#4647—22no—
26#4484—22no—
27#4470—21no—
28#4470—21no—
29#4451—22no—
30#3037—31no—
31#2351—31no—
32#1517—31no—
33#1502—31no—
34#1479—31no—
35#1303—31no—
36#1286D311yescharacteristic, complemented
37C2×#235—21no—
38C22×Dic6D411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42C2311yescharacteristic
42#930—21no—
43#930—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—22no—
48C2×Dic3⋊C4—22no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3—21no—
52#909—42no—
53#909—42no—
54C2×Dic12—22no—
55C2×#204—22no—
56#887—42no—
57#887—42no—
58#875—21no—
59#874C2×C411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869C2×C411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6C2×C412yes—
66C4×Dic6D412yes—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8C2×C412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×#95—21no—
73C3×#95D412yes—
74C3⋊#95—22no—
75C3⋊#95D411yescharacteristic
76C3⋊#95D411yescharacteristic
77#830—21no—
78#821—22no—
79#816—22no—
80C4⋊(C3⋊C8)—21no—
81C4×C3⋊C8—21no—
82C2×#137—61no—
83#532—31no—
84C2×C4×Q8—31no—
85C2×C4×Q8—61no—
86#464—62no—
87#462—32no—
88#460—32no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423C2×D311yescharacteristic
92#417—61no—
93#389—61no—
94#389—62no—
95#382—61no—
96#382C2×D311yescharacteristic
97C4×C4⋊C4—31no—
98C4×C4⋊C4—61no—
99#327—61no—
100#327—61no—
101#327—62no—
102Q8⋊C8—62no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C2×#67—41no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112#235—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×C4×D3—41no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6C22⋊C412yes—
125C2×Dic6D812yes—
126C2×Dic6SD1612yes—
127C3×M16—41no—
128C3×M16—41no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—42no—
143C4⋊Dic3C22⋊C412yes—
144Dic3⋊C4—22no—
145Dic3⋊C4—22no—
146Dic3⋊C4—44no—
147Dic3⋊C4—44no—
148C4×Dic3—22no—
149C4×Dic3—22no—
150C4×Dic3—44no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153C2×C3⋊C8—41no—
154Dic12—44no—
155#204—44no—
156C2×#41—61no—
157C22×Q8—31no—
158#137—122no—
159#137—122no—
160C2×Q16—62no—
161C2×SD16—62no—
162C2×M16—21no—
163C4⋊Q8—31no—
164#127—31no—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×Q8—32no—
169C4×Q8—32no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×Q8—122no—
173C4×Q8—122no—
174C4×D4—61no—
175C4×D4—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—61no—
182C2×C42—61no—
183C2×C42—62no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189C4⋊C8C4×D312yes—
190#103—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C4×C8—21no—
195C4×C8—61no—
196#95—21no—
197#95—31no—
198#95—31no—
199#95—62no—
200#95D1212yes—
201C23×C3#12011yescharacteristic
202C22×D3—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4C2×D811yescharacteristic
215C2×C3×C4C2×SD1611yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4#13711yescharacteristic
218#67—81no—
219#67—81no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232D12—41no—
233D12—41no—
234D12—81no—
235D12—81no—
236C4×D3—81no—
237C4×D3—81no—
238Dic6—22no—
239Dic6—22no—
240Dic6—22no—
241Dic6—41no—
242Dic6—41no—
243Dic6#10214yes—
244C3×C8—42no—
245C3×C8—81no—
246C3⋊C8—81no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—32no—
252C2×Q8—32no—
253C2×Q8—32no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C22×C4—22no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—121no—
268C22×C4C2×D1211yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271Q16—124no—
272SD16—124no—
273M16—41no—
274M16—41no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—121no—
278C4⋊C4—32no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—124no—
287C4⋊C4—124no—
288C4⋊C4—122no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42—62no—
292C42—62no—
293C42—124no—
294C42—121no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42#23511yescharacteristic
298C42D3×D411yescharacteristic
299C42#23911yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3#38611yescharacteristic
303C22×C3#44711yescharacteristic
304C22×C3#44811yescharacteristic
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—82no—
313C3×C4C2×#10211yescharacteristic
314C3×C4#41711yescharacteristic
315C3×C4#48011yescharacteristic
316C3×C4#48111yescharacteristic
317Dic3—24no—
318Dic3—42no—
319Dic3—42no—
320Dic3—81no—
321Dic3—81no—
322C23—121no—
323C23#88211yescharacteristic
324Q8—34no—
325Q8—62no—
326Q8—62no—
327Q8—62no—
328Q8—121no—
329Q8—121no—
330D4—121no—
331D4—121no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4C4×D1211yescharacteristic
356C2×C4D3×C22⋊C411yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4#89311yescharacteristic
359C2×C4#89411yescharacteristic
360C2×C4#89511yescharacteristic
361C2×C4D3×D811yescharacteristic
362C2×C4D3×SD1611yescharacteristic
363C2×C4#91411yescharacteristic
364C2×C4#91811yescharacteristic
365C8—42no—
366C8—81no—
367C8—241no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3#187411yescharacteristic
371C2×C3#190511yescharacteristic
372C2×C3#190611yescharacteristic
373D3—161no—
374C22—21no—
375C22—21no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22#466511yescharacteristic
380C22#477311yescharacteristic
381C22#480611yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—64no—
387C4—82no—
388C4—122no—
389C4—122no—
390C4—241no—
391C4—241no—
392C4D3×#10211yescharacteristic
393C4#480311yescharacteristic
394C4#486311yescharacteristic
395C4#488211yescharacteristic
396C3#918011yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—481no—
400C2#6870711yescharacteristic
401C2#6888011yescharacteristic
402C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9180
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
2. #74157C2semidirect
4. #70766C2semidirect
6. C3×#1286C2semidirect
7. #67715C2semidirect
20. #4849C4semidirect
21. #4848C4semidirect
36. #1286D3semidirect
396. C3#9180semidirect

Maximal subgroups

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

subgroupindex
#790672
#741572
#707892
#707662
#695452
C3×#12862
#677152
#91803