Finite Group Databasebeta

← #10699181

Subgroups — #10699181

Order 768.

440 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 99 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#83213C211yesmaximal, characteristic
2C3×#1997C211yesmaximal, characteristic, complemented
3#71778C211yesmaximal, characteristic, complemented
4#70503C211yesmaximal, characteristic
5#69867C211yesmaximal, characteristic
6#69850C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10501—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5227—22no—
14C2×#835—22no—
15C2×#825—21no—
16C2×#825—21no—
17#5085—21no—
18#5083—21no—
19#5083—21no—
20#5079—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×#343—21no—
25C3×#343—21no—
26#4485—21no—
27#4482—21no—
28#4458—21no—
29#4457—21no—
30#3206—31no—
31#1997D311yescharacteristic, complemented
32#1981—31no—
33#1980—31no—
34#1565—31no—
35#1547—31no—
36#1303—31no—
37C22×C3×Q8D412yes—
38C2×#235—21no—
39C2×C3×M16—21no—
40C2×C3×M16D411yescharacteristic
41C2×C3×M16D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#215—22no—
50C2×#215—22no—
51#930—21no—
52#930—21no—
53C2×C4⋊Dic3—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56#875—21no—
57#874D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#869D411yescharacteristic
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×#97—21no—
64C3×C4×C8—21no—
65#835—22no—
66#835—22no—
67#835—44no—
68C3⋊#95—21no—
69#830—21no—
70#825—42no—
71#825—42no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D411yescharacteristic
75C4⋊(C3⋊C8)D411yescharacteristic
76C4×C3⋊C8—21no—
77#532—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79#492—61no—
80#487—61no—
81#461—61no—
82#461—61no—
83#448—62no—
84#431—31no—
85#423C2×D311yescharacteristic
86C2×C4⋊C8—31no—
87C2×#104—61no—
88C2×#104—61no—
89C2×#103—62no—
90C4×M16C2×D311yescharacteristic
91C4×C4⋊C4—31no—
92#343—21no—
93#343—21no—
94#335—61no—
95#334—61no—
96#332—61no—
97#331—61no—
98C2×C3×Q8—22no—
99C2×C3×Q8—22no—
100C2×C3×Q8—42no—
101C2×C3×Q8—42no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×#67—41no—
107C22×Dic3—41no—
108#235—41no—
109#235—41no—
110#235—41no—
111#235—41no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×C4×D3—41no—
116C2×Dic6—21no—
117C3×M16—21no—
118C3×M16—21no—
119C3×M16—21no—
120C3×M16—41no—
121C3×M16—41no—
122C3×M16SD1612yescomplemented
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134#215—82no—
135#215—82no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—41no—
142C4×Dic3—41no—
143C4×Dic3—41no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—42no—
147C2×C3⋊C8—41no—
148C2×#41—61no—
149C22×Q8D1212yes—
150C2×SD16—62no—
151C2×SD16—62no—
152C2×M16—21no—
153C2×M16#6711yescharacteristic
154C2×M16#6711yescharacteristic
155C22×C8—61no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—31no—
161#127—31no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—61no—
170C2×C42—61no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—31no—
175C4⋊C8—31no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178#104—122no—
179#104—122no—
180#103—62no—
181#103—62no—
182#103—124no—
183C22⋊C8—61no—
184#97—21no—
185C4×C8—21no—
186C4×C8—61no—
187#95—61no—
188C23×C3#12111yescharacteristic
189C22×D3—41no—
190C3×Q8—44no—
191C3×Q8—42no—
192C3×Q8—42no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—82no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12311yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×C3×C4#12411yescharacteristic
204C2×C3×C4C2×SD1611yescharacteristic
205C2×C3×C4#13511yescharacteristic
206C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C3×C4#13711yescharacteristic
208#67—81no—
209#67—81no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214D12—41no—
215D12—41no—
216D12—41no—
217D12—41no—
218D12—81no—
219C4×D3—81no—
220C4×D3—81no—
221Dic6—41no—
222Dic6—41no—
223C3×C8—22no—
224C3×C8—41no—
225C3×C8—81no—
226C3⋊C8—41no—
227C3⋊C8—41no—
228C3⋊C8—81no—
229#41—121no—
230#41—121no—
231#41—121no—
232#41—121no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C22×C4—22no—
243C22×C4—121no—
244C22×C4—121no—
245C22×C4C2×D1211yescharacteristic
246C22×C4#23511yescharacteristic
247C22×C4C2×#6711yescharacteristic
248SD16—242no—
249SD16—242no—
250M16—21no—
251M16—21no—
252M16—21no—
253M16—41no—
254M16—41no—
255M16#21512yescomplemented
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—41no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—122no—
262C2×C8—121no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—62no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C42—121no—
274C42—121no—
275C42D3×D411yescharacteristic
276C42D3×D411yescharacteristic
277C42#23711yescharacteristic
278C42#23711yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3#39411yescharacteristic
282C22×C3#46011yescharacteristic
283C22×C3#46411yescharacteristic
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—82no—
291C3×C4#45311yescharacteristic
292C3×C4#45511yescharacteristic
293C3×C4#46111yescharacteristic
294C3×C4#46311yescharacteristic
295C3×C4#46511yescharacteristic
296C3×C4#46811yescharacteristic
297Dic3—82no—
298Dic3—81no—
299C23—121no—
300C23#93011yescharacteristic
301Q8—44no—
302Q8—42no—
303Q8—42no—
304Q8—121no—
305Q8—121no—
306D4—121no—
307D4—121no—
308D4—121no—
309D4—121no—
310D4—241no—
311D4—241no—
312D4—241no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—82no—
318C2×C4—122no—
319C2×C4—121no—
320C2×C4—241no—
321C2×C4—241no—
322C2×C4—241no—
323C2×C4—241no—
324C2×C4#88211yescharacteristic
325C2×C4#88311yescharacteristic
326C2×C4#88511yescharacteristic
327C2×C4#88611yescharacteristic
328C2×C4#93811yescharacteristic
329C2×C4#93911yescharacteristic
330C2×C4C2×#21511yescharacteristic
331C2×C4#94211yescharacteristic
332C2×C4#94911yescharacteristic
333C2×C4#95011yescharacteristic
334C8—22no—
335C8—41no—
336C8—81no—
337C8—121no—
338C8—121no—
339C8—241no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3#203711yescharacteristic
343C2×C3#204911yescharacteristic
344C2×C3#205511yescharacteristic
345D3—161no—
346C22—21no—
347C22—21no—
348C22—241no—
349C22—241no—
350C22—241no—
351C22#496111yescharacteristic
352C22#502711yescharacteristic
353C22#502811yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—82no—
358C4—242no—
359C4—241no—
360C4#474611yescharacteristic
361C4#474711yescharacteristic
362C4#504811yescharacteristic
363C4#504911yescharacteristic
364C4#505711yescharacteristic
365C4#508311yescharacteristic
366C3#1050111yescharacteristic, complemented, Sylow
367C2—21no—
368C2—21no—
369C2—481no—
370C2#6968011yescharacteristic
371C2#7006611yescharacteristic
372C2#7011011yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#394
prime radical#5477C22
hypercenter#1997D3
hypocenterC3#10501
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. C3×#1997C2semidirect
3. #71778C2semidirect
6. #69850C2semidirect
7. #67715C2semidirect
31. #1997D3semidirect
122. C3×M16SD16semidirect
255. M16#215semidirect
366. C3#10501semidirect

Maximal subgroups

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

subgroupindex
#832132
C3×#19972
#717782
#705032
#698672
#698502
#677152
#105013