Finite Group Databasebeta

← #10697356

Subgroups — #10697356

Order 768.

560 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 107 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#83363C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic, complemented
3#78538C211yesmaximal, characteristic
4C3×#1986C211yesmaximal, characteristic, complemented
5#69490C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic, complemented
8#10206—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#909—22no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5084—21no—
16#5083—21no—
17#5080—21no—
18#5079—21no—
19#5048—22no—
20#5043—22no—
21#5027—21no—
22#5020—22no—
23#5020—21no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4715—22no—
27#4699—21no—
28#4693—21no—
29C3×#339—22no—
30#4529—22no—
31#4451—22no—
32#3246—31no—
33#3229—31no—
34#3152—31no—
35#1988—31no—
36#1986D311yescharacteristic, complemented
37#1502—31no—
38#1283—31no—
39C2×#235—21no—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#215—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C22×C3⋊C8D412yes—
59#909—44no—
60#909—44no—
61C2×Dic12—24no—
62C2×#204—24no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869D411yescharacteristic
71#869D412yes—
72C3×#97—21no—
73C3×C4×C8—21no—
74C3×#95—21no—
75#835—24no—
76#835—22no—
77#830D412yes—
78#809—22no—
79#808—24no—
80#808—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85C2×#137—62no—
86#533C2×D311yescharacteristic
87#532—31no—
88C2×C4⋊Q8—31no—
89#498—61no—
90#497—62no—
91#494—61no—
92#492—61no—
93#487—61no—
94#483—62no—
95#476—61no—
96#475—62no—
97#475—61no—
98#465—62no—
99#461—61no—
100#460—61no—
101#431—31no—
102C2×C4⋊C8—31no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4C2×D311yescharacteristic
105#340—62no—
106#339—22no—
107#327—62no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×#67—41no—
115C22×Dic3—41no—
116#235—42no—
117#235—42no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×C4×D3—41no—
122C2×Dic6—22no—
123C2×Dic6—21no—
124C2×Dic6—42no—
125C2×Dic6—42no—
126C2×Dic6—41no—
127C3×M16—22no—
128C3×M16SD1614yescomplemented
129C2×C3×C8—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42—41no—
140C3×C42—42no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#215—82no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—22no—
151C4⋊Dic3—21no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157Dic12—48no—
158#204—48no—
159C2×#41—61no—
160C22×Q8—61no—
161#137—124no—
162#137—124no—
163C2×Q16—64no—
164C2×SD16—64no—
165C2×SD16—62no—
166C2×M16#6712yes—
167C22×C8—32no—
168C4⋊Q8—32no—
169C4⋊Q8—31no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
173C4⋊Q8D1211yescharacteristic
174C4⋊Q8D1211yescharacteristic
175#127—31no—
176#125—21no—
177#124—61no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180#121—61no—
181#121—61no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×D4—61no—
185C4×D4—61no—
186#117—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C42—21no—
191C2×C42C22×D311yescharacteristic
192#106—62no—
193#106—64no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—62no—
198#103—64no—
199#102—62no—
200C22⋊C8—32no—
201#97—21no—
202C4×C8—21no—
203C4×C8—61no—
204#95—21no—
205C23×C3#12711yescharacteristic
206C22×D3—41no—
207C3×Q8—42no—
208C3×Q8—41no—
209C3×Q8—41no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4C2×SD1612yes—
225C2×C3×C4AGL(1,ℤ/8)12yes—
226#67—81no—
227#67—81no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232D12—42no—
233D12—41no—
234D12—41no—
235D12—81no—
236C4×D3—82no—
237Dic6—44no—
238Dic6—42no—
239Dic6—82no—
240C3×C8—24no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243#41—122no—
244#41—122no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—61no—
248C2×Q8—62no—
249C2×Q8—121no—
250C2×Q8—122no—
251C2×Q8—122no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—41no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—128no—
263SD16—128no—
264SD16—242no—
265M16—22no—
266M16#21514yescomplemented
267C2×C8—22no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—122no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—62no—
280C4⋊C4—61no—
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C42—41no—
289C42—42no—
290C42C2×D1211yescharacteristic
291C42#23511yescharacteristic
292C42D3×D411yescharacteristic
293C42D3×D411yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3#39211yescharacteristic
297C22×C3#49211yescharacteristic
298C22×C3#49611yescharacteristic
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—82no—
306C3×C4—81no—
307C3×C4—81no—
308C3×C4#45012yes—
309C3×C4#45511yescharacteristic
310C3×C4#45611yescharacteristic
311C3×C4#46012yes—
312Dic3—82no—
313Dic3—81no—
314C23—121no—
315C23#94011yescharacteristic
316Q8—42no—
317Q8—41no—
318Q8—41no—
319Q8—124no—
320Q8—122no—
321Q8—242no—
322D4—122no—
323D4—121no—
324D4—121no—
325D4—241no—
326D4—241no—
327D4—241no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4—242no—
341C2×C4#88211yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4#93011yescharacteristic
346C2×C4#93711yescharacteristic
347C2×C4C2×#21512yes—
348C2×C4#94912yes—
349C8—24no—
350C8—122no—
351C8—122no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3#200711yescharacteristic
355C2×C3#200911yescharacteristic
356C2×C3#201411yescharacteristic
357D3—161no—
358C22—21no—
359C22—21no—
360C22—241no—
361C22—241no—
362C22—241no—
363C22#496011yescharacteristic
364C22#505811yescharacteristic
365C22#508511yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—82no—
370C4—81no—
371C4—81no—
372C4—241no—
373C4—242no—
374C4#482511yescharacteristic
375C4#482611yescharacteristic
376C4#502712yes—
377C4#504612yes—
378C3#1020611yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—481no—
382C2#6968811yescharacteristic
383C2#7039311yescharacteristic
384C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1986D3
hypocenterC3#10206
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. #83323C2semidirect
4. C3×#1986C2semidirect
5. #69490C2semidirect
7. #67668C2semidirect
36. #1986D3semidirect
128. C3×M16SD16semidirect
266. M16#215semidirect
378. C3#10206semidirect

Maximal subgroups

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

subgroupindex
#833632
#833232
#785382
C3×#19862
#694902
#694692
#676682
#102063