Finite Group Databasebeta

← #11782915

Subgroups — #11782915

Order 1152.

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

normal subgroups only

This group has 565 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#10592772C211yesmaximal, characteristic, complemented
2C3×#5076C211yesmaximal, characteristic, complemented
3C3×#5073C211yesmaximal, characteristic, complemented
4#10590350C211yesmaximal, characteristic, complemented
5#10590281C211yesmaximal, characteristic, complemented
6#10590277C211yesmaximal, characteristic, complemented
7#10589078C211yesmaximal, characteristic, complemented
8#71279—31nomaximal
9#71262—31nomaximal
10C32×#127C2211yescharacteristic
11C2×#3753—22no—
12#64083C2211yescharacteristic
13C2×C3×#214—22no—
14C2×C3×#213—22no—
15C3×#874C2211yescharacteristic
16C3×#869C2211yescharacteristic
17C2×#3714—22no—
18C4×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63564—22no—
22#63562—22no—
23C3×#493—31no—
24C3×#492—31no—
25#5088—31no—
26#5088—31no—
27#5076D311yescharacteristic, complemented
28#5073—31no—
29#5073—31no—
30#5073—31no—
31#5073—61no—
32#5073D311yescharacteristic, complemented
33#5070—31no—
34#4695—31no—
35#4694—31no—
36#4455—31no—
37#4454—31no—
38C2×C32×D4—21no—
39C2×C32×D4—21no—
40C2×C32×D4—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×Dic6—21no—
48C32×C42C2311yescharacteristic
49#3753—44no—
50C2×C3⋊(C3⋊C8)D412yes—
51C3×#214—44no—
52C3×#213—44no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3—21no—
55C2×C3×C3⋊C8D412yes—
56C2×C3×C3⋊C8D412yes—
57#3714—44no—
58#1721—91noSylow
59C2×C3×SD16—62no—
60C2×C3×D8—62no—
61C3×C4⋊Q8—31no—
62C3×#127—21no—
63C3×#127—31no—
64C3×#127C2×D311yescharacteristic
65C3×#127C2×D311yescharacteristic
66C2×#214—22no—
67C2×#213—22no—
68C2×#213—62no—
69C2×#213—62no—
70C2×#213—64no—
71C2×#213—62no—
72C2×D24—62no—
73#874—31no—
74#874—31no—
75#874—61no—
76#874C2×D311yescharacteristic
77#869C2×D311yescharacteristic
78C3×C4×C8—31no—
79C3×C4×C8—31no—
80#832—62no—
81#821—62no—
82#809—62no—
83#809—62no—
84C4×C3⋊C8—31no—
85C4×C3⋊C8—31no—
86C4×C3⋊C8—61no—
87C4×C3⋊C8C2×D311yescharacteristic
88C4×C3⋊C8C2×D311yescharacteristic
89C23×C32—41no—
90C22×C3⋊D3—41no—
91C22×C3×D3—41no—
92C32×D4—42no—
93C32×D4—42no—
94C32×D4—42no—
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C4⋊(C3⋊D3)—42no—
99C4⋊(C3⋊D3)—42no—
100C4⋊(C3⋊D3)—42no—
101C2×C3×Dic3—41no—
102C3×D12—42no—
103C3×D12—42no—
104C3×D12—42no—
105C3×Dic6—42no—
106C3⋊(C3⋊C8)—22no—
107C3×C3⋊C8—22no—
108C3×C3⋊C8D814yescomplemented
109#493—31no—
110#493—91no—
111#493—91no—
112#492—31no—
113#486—91no—
114#486—91no—
115#322—91no—
116C2×C3×Q8—61no—
117C2×C3×D4—22no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—61no—
127C2×C3×D4—61no—
128C2×C3×D4—61no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—62no—
133C2×D12—61no—
134C2×D12—61no—
135C2×D12—61no—
136C2×D12—61no—
137C2×D12—61no—
138C2×D12—61no—
139C2×D12—121no—
140C2×D12—121no—
141C2×Dic6—21no—
142C3×SD16—124no—
143C3×D8—124no—
144C2×C3×C8—32no—
145C2×C3×C8—32no—
146C3×C4⋊C4—61no—
147C3×C4⋊C4—61no—
148C3×C42—21no—
149C3×C42C22×D311yescharacteristic
150C3×C42C22×D311yescharacteristic
151#214—44no—
152#213—44no—
153#213—128no—
154#213—124no—
155#213—124no—
156#213—124no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—62no—
162C2×C3⋊C8#6712yes—
163C2×C3⋊C8#6712yes—
164D24—124no—
165C22×C32—81no—
166C22×C32—81no—
167C22×C32—81no—
168C22×C32#12711yescharacteristic
169C2×C3⋊D3—81no—
170C2×C3⋊D3—81no—
171C2×C3⋊D3—81no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C2×C3×D3—81no—
175C32×C4C2×D812yes—
176C32×C4AGL(1,ℤ/8)12yes—
177C32×C4AGL(1,ℤ/8)12yes—
178C3×Dic3—81no—
179C2×SD16—62no—
180C2×D8—62no—
181C2×D8—182no—
182C2×D8—182no—
183C4⋊Q8—31no—
184#127—31no—
185#127—91no—
186#127D3211yescharacteristic
187#102—182no—
188#102—182no—
189C4×C8—31no—
190C4×C8—31no—
191C4×C8—91no—
192C23×C3—42no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—121no—
196C22×D3—41no—
197C22×D3—122no—
198C22×D3—121no—
199C22×D3—121no—
200C3×Q8—122no—
201C3×D4—44no—
202C3×D4—42no—
203C3×D4—42no—
204C3×D4—42no—
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—42no—
208C3×D4—82no—
209C3×D4—82no—
210C3×D4—122no—
211C3×D4—122no—
212C3×D4—122no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—121no—
217C2×C3×C4D3×D411yescharacteristic
218C2×C3×C4D3×D411yescharacteristic
219C2×C3×C4D3×D411yescharacteristic
220C2×C3×C4D3×D411yescharacteristic
221C2×C3×C4C2×#6711yescharacteristic
222C2×C3×C4C2×#6711yescharacteristic
223C2×Dic3—41no—
224D12—42no—
225D12—42no—
226D12—42no—
227D12—124no—
228D12—122no—
229D12—122no—
230D12—122no—
231D12—122no—
232D12—122no—
233D12—122no—
234D12—242no—
235D12—242no—
236Dic6—42no—
237C3×C8—34no—
238C3×C8—62no—
239C3⋊C8—22no—
240C3⋊C8—64no—
241C3⋊C8—62no—
242C3⋊C8—62no—
243C3⋊C8#21314yescomplemented
244C2×C32—161no—
245C2×C32#49311yescharacteristic
246C2×C32#49611yescharacteristic
247C2×C32#49611yescharacteristic
248C3⋊D3—161no—
249C3×D3—161no—
250C2×Q8—61no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—181no—
258C2×D4—181no—
259C2×D4—181no—
260SD16—124no—
261D8—124no—
262D8—364no—
263D8—364no—
264C2×C8—32no—
265C2×C8—32no—
266C2×C8—92no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C42C2×D3211yescharacteristic
270C22×C3—21no—
271C22×C3—82no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3—241no—
281C22×C3—241no—
282C22×C3—241no—
283C22×C3#94011yescharacteristic
284C22×C3#94011yescharacteristic
285C2×D3—81no—
286C2×D3—81no—
287C2×D3—81no—
288C2×D3—242no—
289C2×D3—242no—
290C2×D3—242no—
291C2×D3—241no—
292C2×D3—241no—
293C2×D3—241no—
294C2×D3—241no—
295C2×D3—241no—
296C2×D3—241no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—241no—
301C3×C4D3×D812yes—
302C3×C4#91112yes—
303C3×C4#91412yes—
304C3×C4#91412yes—
305C3×C4C2×#21312yes—
306C3×C4#93212yes—
307Dic3—81no—
308C32#172111yescharacteristic, complemented, Sylow
309C23—41no—
310C23—121no—
311C23—361no—
312Q8—122no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—122no—
317D4—122no—
318D4—122no—
319D4—362no—
320D4—362no—
321D4—362no—
322C2×C4—121no—
323C2×C4D3×#6711yescharacteristic
324C2×C4D3×#6711yescharacteristic
325C2×C4#381111yescharacteristic
326C8—34no—
327C8—62no—
328C8—182no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—162no—
333C2×C3—161no—
334C2×C3—161no—
335C2×C3—481no—
336C2×C3#507311yescharacteristic
337C2×C3#507411yescharacteristic
338C2×C3#515111yescharacteristic
339C2×C3#515411yescharacteristic
340C2×C3#517611yescharacteristic
341C2×C3#517611yescharacteristic
342D3—161no—
343D3—482no—
344D3—481no—
345D3—481no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—241no—
350C22—241no—
351C22—241no—
352C22—721no—
353C22—721no—
354C22—721no—
355C22#6397911yescharacteristic
356C4—241no—
357C4D3×#21312yes—
358C4#6393412yes—
359C4#6393712yes—
360C3—21no—
361C3#7126211yescharacteristic, complemented
362C3#7127911yescharacteristic, complemented
363C2—161no—
364C2—481no—
365C2—1441no—
366C2#1059147211yescharacteristic
367C2#1059147511yescharacteristic
368C2#1059148211yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#127C22
socleC22×C32#127
hypercenter#127D32
hypocenterC32#1721
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #10592772C2semidirect
2. C3×#5076C2semidirect
3. C3×#5073C2semidirect
4. #10590350C2semidirect
5. #10590281C2semidirect
6. #10590277C2semidirect
7. #10589078C2semidirect
27. #5076D3semidirect
32. #5073D3semidirect
108. C3×C3⋊C8D8semidirect
243. C3⋊C8#213semidirect
308. C32#1721semidirect
361. C3#71262semidirect
362. C3#71279semidirect

Maximal subgroups

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

subgroupindex
#105927722
C3×#50762
C3×#50732
#105903502
#105902812
#105902772
#105890782
#712793
#712623