Finite Group Databasebeta

← #11782923

Subgroups — #11782923

Order 1152.

543 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 102 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#10592787C211yesmaximal, characteristic, complemented
2C3×#5088C211yesmaximal, characteristic, complemented
3C3×#5085C211yesmaximal, characteristic, complemented
4#10590352C211yesmaximal, characteristic, complemented
5#10590350C211yesmaximal, characteristic, complemented
6#10590350C211yesmaximal, characteristic, complemented
7#10589078C211yesmaximal, characteristic, complemented
8#71262—31nomaximal
9#71194—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64083C2211yescharacteristic
12C2×C3×#215—22no—
13C3×#874C2211yescharacteristic
14C3×#874C2211yescharacteristic
15C2×#3714—22no—
16C2×#3714—22no—
17C2×#3713—22no—
18#63636—22no—
19C4×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×#809—22no—
21C3×C4×C3⋊C8C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23C3×#492—31no—
24C3×#486—31no—
25#5088—31no—
26#5088—31no—
27#5088—61no—
28#5088D311yescharacteristic, complemented
29#5085D311yescharacteristic, complemented
30#5073—31no—
31#5073—31no—
32#5073—31no—
33#5073—31no—
34#4695—31no—
35#4695—31no—
36#4455—31no—
37#4454—31no—
38C2×C32×Q8—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C42C2311yescharacteristic
51C2×C3⋊(C3⋊C8)D412yes—
52C3×#215—44no—
53C2×C3×C3⋊C8D412yes—
54C2×C3×C3⋊C8D412yes—
55#3714—44no—
56#3714—44no—
57#3713—44no—
58#1721—91noSylow
59C2×C3×SD16—62no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×#127—31no—
64C3×#127—31no—
65C2×#215—22no—
66C2×#213—62no—
67C2×#213—62no—
68C2×#213—62no—
69C2×#213—62no—
70C2×D24—62no—
71C2×D24—62no—
72#874—31no—
73#874—31no—
74#874—61no—
75#874C2×D311yescharacteristic
76#874C2×D311yescharacteristic
77C3×#102—62no—
78C3×C4×C8—31no—
79C3×C4×C8—31no—
80#809—22no—
81#809—62no—
82#809—62no—
83#809—64no—
84C4×C3⋊C8—31no—
85C4×C3⋊C8—31no—
86C4×C3⋊C8—61no—
87C4×C3⋊C8C2×D311yescharacteristic
88C4×C3⋊C8C2×D311yescharacteristic
89C22×C3⋊D3—41no—
90C22×C3×D3—41no—
91C22×C3×D3—41no—
92C32×Q8—42no—
93C2×C32×C4—41no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C4⋊(C3⋊D3)—42no—
98C4⋊(C3⋊D3)—42no—
99C4⋊(C3⋊D3)—42no—
100C3×D12—42no—
101C3×D12—42no—
102C3×D12—42no—
103C3×D12—42no—
104C3×D12—42no—
105C3×D12—42no—
106C3⋊(C3⋊C8)—22no—
107C3×C3⋊C8—22no—
108C3×C3⋊C8D814yescomplemented
109#493—91no—
110#493—91no—
111#493—91no—
112#492—31no—
113#486—31no—
114#486—91no—
115#322—91no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—41no—
119C2×C3×D4—61no—
120C2×C3×D4—61no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—62no—
132C2×D12—61no—
133C2×D12—61no—
134C2×D12—61no—
135C2×D12—61no—
136C2×D12—61no—
137C2×D12—61no—
138C2×D12—121no—
139C2×D12—121no—
140C3×SD16—124no—
141C2×C3×C8—32no—
142C2×C3×C8—32no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C42—21no—
150C3×C42C22×D311yescharacteristic
151C3×C42C22×D311yescharacteristic
152#215—44no—
153#213—124no—
154#213—124no—
155#213—124no—
156#213—124no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—62no—
160C2×C3⋊C8#6712yes—
161C2×C3⋊C8#6712yes—
162D24—124no—
163D24—124no—
164C22×C32#12711yescharacteristic
165C2×C3⋊D3—81no—
166C2×C3⋊D3—81no—
167C2×C3⋊D3—81no—
168C2×C3×D3—81no—
169C2×C3×D3—81no—
170C2×C3×D3—81no—
171C2×C3×D3—81no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—81no—
175C32×C4C2×D812yes—
176C32×C4AGL(1,ℤ/8)12yes—
177C32×C4AGL(1,ℤ/8)12yes—
178C2×SD16—62no—
179C2×D8—182no—
180C2×D8—182no—
181C2×D8—182no—
182C4⋊Q8D3211yescharacteristic
183#127—31no—
184#127—31no—
185#127—91no—
186#102—62no—
187#102—182no—
188C4×C8—31no—
189C4×C8—31no—
190C4×C8—91no—
191C23×C3—121no—
192C23×C3—121no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—122no—
196C22×D3—121no—
197C22×D3—121no—
198C3×Q8—42no—
199C3×Q8—42no—
200C3×Q8—82no—
201C3×D4—122no—
202C3×D4—122no—
203C3×D4—122no—
204C3×D4—122no—
205C3×D4—122no—
206C3×D4—122no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4D3×D411yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4D3×D411yescharacteristic
216C2×C3×C4D3×D411yescharacteristic
217C2×C3×C4C2×#6711yescharacteristic
218C2×C3×C4C2×#6711yescharacteristic
219D12—42no—
220D12—42no—
221D12—42no—
222D12—42no—
223D12—42no—
224D12—42no—
225D12—124no—
226D12—122no—
227D12—122no—
228D12—122no—
229D12—122no—
230D12—122no—
231D12—122no—
232D12—242no—
233D12—242no—
234C3×C8—34no—
235C3×C8—62no—
236C3⋊C8—22no—
237C3⋊C8—62no—
238C3⋊C8—62no—
239C3⋊C8—122no—
240C3⋊C8#21314yescomplemented
241C2×C32#49311yescharacteristic
242C2×C32#49611yescharacteristic
243C2×C32#49611yescharacteristic
244C3⋊D3—161no—
245C3×D3—161no—
246C3×D3—161no—
247C2×Q8—21no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—181no—
255C2×D4—181no—
256C2×D4—181no—
257SD16—124no—
258D8—364no—
259D8—364no—
260D8—364no—
261C2×C8—32no—
262C2×C8—32no—
263C2×C8—92no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C42C2×D3211yescharacteristic
267C22×C3—21no—
268C22×C3—241no—
269C22×C3—241no—
270C22×C3—241no—
271C22×C3—241no—
272C22×C3—241no—
273C22×C3—241no—
274C22×C3#94011yescharacteristic
275C22×C3#94011yescharacteristic
276C2×D3—81no—
277C2×D3—81no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C2×D3—242no—
283C2×D3—242no—
284C2×D3—242no—
285C2×D3—241no—
286C2×D3—241no—
287C2×D3—241no—
288C2×D3—241no—
289C2×D3—241no—
290C2×D3—241no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—82no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4D3×D812yes—
298C3×C4#91412yes—
299C3×C4#91412yes—
300C3×C4#91412yes—
301C3×C4C2×#21312yes—
302C3×C4#94912yes—
303C32#172111yescharacteristic, complemented, Sylow
304C23—121no—
305C23—121no—
306C23—361no—
307Q8—42no—
308D4—122no—
309D4—122no—
310D4—122no—
311D4—122no—
312D4—122no—
313D4—122no—
314D4—362no—
315D4—362no—
316D4—362no—
317C2×C4—41no—
318C2×C4D3×#6711yescharacteristic
319C2×C4D3×#6711yescharacteristic
320C2×C4#381111yescharacteristic
321C8—34no—
322C8—62no—
323C8—182no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—481no—
328C2×C3—481no—
329C2×C3#505811yescharacteristic
330C2×C3#507311yescharacteristic
331C2×C3#515111yescharacteristic
332C2×C3#517611yescharacteristic
333C2×C3#517611yescharacteristic
334C2×C3#517611yescharacteristic
335D3—161no—
336D3—161no—
337D3—482no—
338D3—481no—
339D3—481no—
340C22—241no—
341C22—241no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22—241no—
346C22—721no—
347C22—721no—
348C22—721no—
349C22#6397911yescharacteristic
350C4—81no—
351C4D3×#21312yes—
352C4#6393912yes—
353C4#6394112yes—
354C3—21no—
355C3#7119411yescharacteristic, complemented
356C3#7126211yescharacteristic, complemented
357C2—481no—
358C2—481no—
359C2—1441no—
360C2#1059148211yescharacteristic
361C2#1059161911yescharacteristic
362C2#1059162111yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
hypercenterC4⋊Q8D32
hypocenterC32#1721
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #10592787C2semidirect
2. C3×#5088C2semidirect
3. C3×#5085C2semidirect
4. #10590352C2semidirect
5. #10590350C2semidirect
6. #10590350C2semidirect
7. #10589078C2semidirect
28. #5088D3semidirect
29. #5085D3semidirect
108. C3×C3⋊C8D8semidirect
240. C3⋊C8#213semidirect
303. C32#1721semidirect
355. C3#71194semidirect
356. C3#71262semidirect

Maximal subgroups

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

subgroupindex
#105927872
C3×#50882
C3×#50852
#105903522
#105903502
#105890782
#712623
#711943