Finite Group Databasebeta

← #11781309

Subgroups — #11781309

Order 1152.

384 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 84 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#10592547C211yesmaximal, characteristic
2#10590284C211yesmaximal, characteristic, complemented
3#10590281C211yesmaximal, characteristic
4#10590254C211yesmaximal, characteristic, complemented
5C3×#4481C211yesmaximal, characteristic, complemented
6C3×#4480C211yesmaximal, characteristic, complemented
7#10589091C211yesmaximal, characteristic
8#68548—31nomaximal
9#68510—31nomaximal
10C2×#3744—21no—
11#64083C2211yescharacteristic
12C4×#634C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15#63648—21no—
16C3×C4⋊(C3⋊C8)C2211yescharacteristic
17C3×C4⋊(C3⋊C8)C2211yescharacteristic
18C3×C4×C3⋊C8C2211yescharacteristic
19#63575—21no—
20#63572—21no—
21#63567—21no—
22#63564—21no—
23#63564—21no—
24#63553—21no—
25#5091—31no—
26#5088—31no—
27#4850—31no—
28#4848—31no—
29#4848—31no—
30#4848—61no—
31#4830—31no—
32#4694—31no—
33C8×Dic6—31no—
34C3×#334—31no—
35C3×#333—31no—
36#4485—31no—
37#4485—31no—
38#4481D311yescharacteristic, complemented
39#4480D311yescharacteristic, complemented
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C2×#634D411yescharacteristic
44C2×C3×Dic6—21no—
45C2×C32×C8—21no—
46C32×C42C2311yescharacteristic
47C4⋊(C3⋊Dic3)D411yescharacteristic
48#3750—21no—
49C4×C3⋊Dic3—21no—
50#3744—42no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57#1678—91noSylow
58C3×C4⋊Q8—31no—
59C2×#204—61no—
60C2×#204—61no—
61C2×#204—62no—
62#874—31no—
63#874—31no—
64#874—61no—
65#869C2×D311yescharacteristic
66C4×Dic6—31no—
67C4×Dic6—31no—
68C4×Dic6—61no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8—31no—
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4⋊C8C2×D311yescharacteristic
74C3×C4×C8—31no—
75#821—61no—
76#821—61no—
77#821—61no—
78#821—61no—
79#821—62no—
80#818—61no—
81#816—61no—
82Dic3⋊C8—61no—
83Dic3⋊C8—61no—
84C8×Dic3—61no—
85C8×Dic3—61no—
86#810—61no—
87#809—61no—
88#809—61no—
89#807—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3⋊D3—41no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4111yescharacteristic
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C4⋊(C3⋊D3)—41no—
100C4⋊(C3⋊D3)—41no—
101C4⋊(C3⋊D3)—41no—
102C4⋊(C3⋊D3)—41no—
103C4⋊(C3⋊D3)—81no—
104#634—21no—
105#634SD1612yes—
106C2×C3×Dic3—41no—
107C3×Dic6—81no—
108C32×C8—41no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8—41no—
113#486—91no—
114#477—91no—
115#460—91no—
116C8×Q8—91no—
117#334—31no—
118#333—31no—
119#331—91no—
120C2×C3×Q8—61no—
121C2×D12—62no—
122C2×D12—61no—
123C2×D12—61no—
124C2×D12—61no—
125C2×D12—61no—
126C2×D12—61no—
127C2×D12—61no—
128C2×D12—121no—
129C2×D12—121no—
130C2×Dic6—21no—
131C2×Dic6—31no—
132C2×Dic6—31no—
133C2×Dic6—61no—
134C2×C3×C8—22no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C2×C3×C8—31no—
138C2×C3×C8—31no—
139C2×C3×C8—61no—
140C2×C3×C8—61no—
141C3×C4⋊C4—61no—
142C3×C4⋊C4—61no—
143C3×C42—21no—
144C3×C42C22×D311yescharacteristic
145C3×C42C22×D311yescharacteristic
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—61no—
151Dic3⋊C4—61no—
152Dic3⋊C4—61no—
153Dic3⋊C4—121no—
154C4×Dic3—61no—
155C4×Dic3—61no—
156C4×Dic3—121no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8D1211yescharacteristic
160C2×C3⋊C8D1211yescharacteristic
161#204—122no—
162#204—122no—
163#204—124no—
164C22×C32#12111yescharacteristic
165C2×C3⋊D3—81no—
166C2×C3⋊D3—81no—
167C2×C3⋊D3—81no—
168C32×C4—21no—
169C32×C4C2×SD1611yescharacteristic
170C32×C4#13511yescharacteristic
171C32×C4#13511yescharacteristic
172C32×C4#13711yescharacteristic
173C3⋊Dic3—22no—
174C3⋊Dic3—41no—
175C3×Dic3—81no—
176C2×SD16—181no—
177C4⋊Q8—31no—
178#127—91no—
179C4×Q8—91no—
180#107—181no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—181no—
184C4⋊C8D3211yescharacteristic
185#103—181no—
186#102—181no—
187#102—181no—
188#102—181no—
189C4×C8—31no—
190C4×C8—181no—
191C22×D3—122no—
192C22×D3—121no—
193C22×D3—121no—
194C3×Q8—241no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—121no—
199C2×C3×C4C2×D1211yescharacteristic
200C2×C3×C4#23511yescharacteristic
201C2×C3×C4D3×D411yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4#23911yescharacteristic
205C2×Dic3—41no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—121no—
211C2×Dic3—121no—
212D12—122no—
213D12—122no—
214D12—121no—
215D12—121no—
216D12—121no—
217D12—121no—
218D12—121no—
219D12—121no—
220D12—121no—
221D12—121no—
222D12—241no—
223D12—241no—
224D12—241no—
225D12—241no—
226D12—242no—
227Dic6—32no—
228Dic6—32no—
229Dic6—62no—
230Dic6—61no—
231Dic6—61no—
232Dic6—81no—
233Dic6—121no—
234C3×C8—42no—
235C3×C8—41no—
236C3×C8—41no—
237C3×C8—61no—
238C3×C8—61no—
239C3×C8—121no—
240C3×C8—121no—
241C3⋊C8—21no—
242C3⋊C8—21no—
243C3⋊C8—41no—
244C3⋊C8—41no—
245C2×C32#46411yescharacteristic
246C2×C32#46511yescharacteristic
247C2×C32#47211yescharacteristic
248C3⋊D3—161no—
249C2×Q8—61no—
250C2×Q8—91no—
251C2×D4—181no—
252C2×D4—181no—
253C2×D4—181no—
254SD16—362no—
255C2×C8—21no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—61no—
259C2×C8—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—91no—
263C4⋊C4—181no—
264C42—181no—
265C42C2×D3211yescharacteristic
266C22×C3—21no—
267C22×C3#88411yescharacteristic
268C22×C3#89511yescharacteristic
269C2×D3—242no—
270C2×D3—242no—
271C2×D3—242no—
272C2×D3—241no—
273C2×D3—241no—
274C2×D3—241no—
275C2×D3—241no—
276C2×D3—241no—
277C2×D3—241no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—241no—
286C3×C4#90411yescharacteristic
287C3×C4#90911yescharacteristic
288C3×C4D3×SD1611yescharacteristic
289C3×C4D3×SD1611yescharacteristic
290C3×C4#91611yescharacteristic
291C3×C4#91611yescharacteristic
292C3×C4#91811yescharacteristic
293C3×C4#91911yescharacteristic
294Dic3—62no—
295Dic3—62no—
296Dic3—81no—
297Dic3—122no—
298Dic3—121no—
299Dic3—121no—
300Dic3—241no—
301C32#167811yescharacteristic, complemented, Sylow
302C23—361no—
303Q8—92no—
304Q8—181no—
305Q8—241no—
306D4—361no—
307D4—361no—
308D4—361no—
309D4—361no—
310D4—721no—
311C2×C4—121no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4#379911yescharacteristic
315C2×C4D3×D1211yescharacteristic
316C2×C4#381111yescharacteristic
317C8—41no—
318C8—61no—
319C8—61no—
320C8—121no—
321C8—121no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3#474911yescharacteristic
326C2×C3#479811yescharacteristic
327C2×C3#480711yescharacteristic
328C2×C3#484511yescharacteristic
329C2×C3#486411yescharacteristic
330C2×C3#489711yescharacteristic
331D3—482no—
332D3—481no—
333D3—481no—
334C22—721no—
335C22—721no—
336C22—721no—
337C22#6391011yescharacteristic
338C4—21no—
339C4—182no—
340C4—241no—
341C4—361no—
342C4#6380111yescharacteristic
343C4#6380611yescharacteristic
344C4#6393011yescharacteristic
345C4#6395011yescharacteristic
346C3—21no—
347C3#6851011yescharacteristic, complemented
348C3#6854811yescharacteristic, complemented
349C2—1441no—
350C2#1059052311yescharacteristic
351C2#1059081311yescharacteristic
352C2#1059098711yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radical#64083C22
hypercenterC4⋊C8D32
hypocenterC32#1678
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10590284C2semidirect
4. #10590254C2semidirect
5. C3×#4481C2semidirect
6. C3×#4480C2semidirect
38. #4481D3semidirect
39. #4480D3semidirect
301. C32#1678semidirect
347. C3#68510semidirect
348. C3#68548semidirect

Maximal subgroups

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

subgroupindex
#105925472
#105902842
#105902812
#105902542
C3×#44812
C3×#44802
#105890912
#685483
#685103