Finite Group Databasebeta

← #11777905

Subgroups — #11777905

Order 1152.

385 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 96 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#10592537C211yesmaximal, characteristic
2C4×#3717C211yesmaximal, characteristic
3#10590259C211yesmaximal, characteristic, complemented
4C3×#4484C211yesmaximal, characteristic, complemented
5C3×#4479C211yesmaximal, characteristic, complemented
6#10589082C211yesmaximal, characteristic
7#10589078C211yesmaximal, characteristic, complemented
8#68565—31nomaximal
9#68226—31nomaximal
10C2×#3743—21no—
11C4×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×C4×Dic6C2211yescharacteristic
13C2×#3717C411yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15#63647—21no—
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×C4⋊(C3⋊C8)C2211yescharacteristic
18C3×C4×C3⋊C8C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63573C411yescharacteristic
21Dic3⋊(C3⋊C8)—21no—
22#63566C411yescharacteristic
23#63564C411yescharacteristic
24#63554—21no—
25C4×#215—31no—
26#5024—31no—
27#4838—31no—
28#4838—31no—
29#4838—61no—
30#4831—31no—
31C4×#204—31no—
32C3×#335—31no—
33C3×Q8⋊C8—31no—
34#4484D311yescharacteristic, complemented
35#4483—31no—
36#4479D311yescharacteristic, complemented
37#4459—31no—
38#4455—31no—
39#4454—31no—
40C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×C3×Dic6C2×C411yescharacteristic
43C2×C32×C8—21no—
44C32×C42C2311yescharacteristic
45#3752—21no—
46C4⋊(C3⋊Dic3)C2×C411yescharacteristic
47C2×C3⋊(C3⋊C8)D411yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49#3743—42no—
50C3×C4⋊Dic3C2×C411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8C2×C411yescharacteristic
56C2×C3×C3⋊C8C2×C411yescharacteristic
57#3717—22no—
58#1592—91noSylow
59C3×C4×Q8—31no—
60C2×#215—31no—
61C2×#204—31no—
62C2×ΓL(1,𝔽25)—61no—
63C2×ΓL(1,𝔽25)—61no—
64C2×ΓL(1,𝔽25)—62no—
65C4×D12—31no—
66C4×D12—31no—
67C4×D12—61no—
68C4×Dic6C2×D311yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8C2×D311yescharacteristic
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4×C8—31no—
74C3×C4×C8—31no—
75#835—31no—
76#821—31no—
77(C2×D3)⋊C8—61no—
78(C2×D3)⋊C8—61no—
79(C2×D3)⋊C8—62no—
80#817—31no—
81#816—31no—
82#815—61no—
83Dic3⋊C8—61no—
84#809—31no—
85#808—31no—
86C4⋊(C3⋊C8)—61no—
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88#803—61no—
89C4×C3⋊C8—31no—
90C4×C3⋊C8—31no—
91C4×C3⋊C8—61no—
92C4×C3⋊C8C2×D311yescharacteristic
93C4×C3⋊C8C2×D311yescharacteristic
94C22×C3⋊D3—21no—
95C2×C32×C4C22×C411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4#4111yescharacteristic
98C2×C3⋊Dic3—21no—
99C4⋊(C3⋊D3)—21no—
100C4⋊(C3⋊D3)—21no—
101C4×C3⋊D3—41no—
102C4×C3⋊D3—41no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C3×Dic6—21no—
106C3×Dic6—21no—
107C32×C8—41no—
108C3⋊(C3⋊C8)—21no—
109C3⋊(C3⋊C8)—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8M1612yescomplemented
114#446—91no—
115C4×SD16—91no—
116#436—91no—
117#335—31no—
118Q8⋊C8—31no—
119D4⋊C8—91no—
120#322—91no—
121C2×C3×Q8—31no—
122C2×D12—31no—
123C2×D12—31no—
124C2×D12—61no—
125C2×C4×D3—62no—
126C2×C4×D3—61no—
127C2×C4×D3—61no—
128C2×Dic6C4×D311yescharacteristic
129C2×C3×C8—22no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—61no—
135C2×C3×C8—61no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—61no—
138C3×C42—21no—
139C3×C42—61no—
140C3×C42C22×D311yescharacteristic
141C3×C42C22×D311yescharacteristic
142#215—62no—
143(C2×D3)⋊C4—62no—
144(C2×D3)⋊C4—61no—
145(C2×D3)⋊C4—61no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—61no—
149C4⋊Dic3C4×D311yescharacteristic
150Dic3⋊C4—21no—
151C4×Dic3—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—61no—
160C2×C3⋊C8C4×D311yescharacteristic
161C2×C3⋊C8C4×D311yescharacteristic
162#204—62no—
163ΓL(1,𝔽25)—124no—
164ΓL(1,𝔽25)—122no—
165ΓL(1,𝔽25)—122no—
166C22×C32C4×D411yescharacteristic
167C2×C3⋊D3—41no—
168C2×C3⋊D3—41no—
169C2×C3⋊D3—41no—
170C32×C4—21no—
171C32×C4C2×M1611yescharacteristic
172C32×C4#13111yescharacteristic
173C32×C4AGL(1,ℤ/8)11yescharacteristic
174C32×C4#13711yescharacteristic
175C3⋊Dic3—41no—
176C3×Dic3—41no—
177C3×Dic3—41no—
178C2×SD16—91no—
179C2×M16—181no—
180C4×Q8—31no—
181C4×D4—91no—
182#106—91no—
183C4⋊C8—31no—
184C4⋊C8—181no—
185C4⋊C8D3211yescharacteristic
186#103—91no—
187#102—91no—
188C22⋊C8—181no—
189#97—181no—
190C4×C8—31no—
191C4×C8—31no—
192C4×C8—91no—
193C22×D3—62no—
194C22×D3—61no—
195C22×D3—61no—
196C3×Q8—61no—
197C3×Q8—61no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4C2×C4×D311yescharacteristic
204C2×C3×C4C2×C4×D311yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4#23711yescharacteristic
208C2×C3×C4#23911yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—62no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214D12—61no—
215D12—61no—
216D12—61no—
217D12—61no—
218D12—121no—
219D12—121no—
220C4×D3—122no—
221C4×D3—122no—
222C4×D3—121no—
223C4×D3—121no—
224C4×D3—121no—
225C4×D3—121no—
226Dic6—21no—
227Dic6—21no—
228C3×C8—32no—
229C3×C8—42no—
230C3×C8—41no—
231C3×C8—41no—
232C3×C8—61no—
233C3×C8—121no—
234C3×C8—121no—
235C3⋊C8—21no—
236C3⋊C8—41no—
237C3⋊C8—41no—
238C3⋊C8—62no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C3⋊C8—61no—
242C3⋊C8—61no—
243C3⋊C8—121no—
244C3⋊C8ΓL(1,𝔽25)12yescomplemented
245C2×C32#43611yescharacteristic
246C2×C32#44011yescharacteristic
247C2×C32#44411yescharacteristic
248C3⋊D3—81no—
249C2×Q8—31no—
250C2×D4—91no—
251C22×C4—181no—
252SD16—182no—
253M16—362no—
254C2×C8—21no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—61no—
258C2×C8—61no—
259C2×C8—91no—
260C2×C8—91no—
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—91no—
264C22⋊C4—181no—
265C42—61no—
266C42C2×D3211yescharacteristic
267C22×C3—21no—
268C22×C3#67⋊C411yescharacteristic
269C22×C3#89311yescharacteristic
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—122no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—121no—
276C2×D3—121no—
277C2×D3—121no—
278C2×D3—121no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—121no—
287C3×C4—121no—
288C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
289C3×C4#90111yescharacteristic
290C3×C4D3×M1611yescharacteristic
291C3×C4#90711yescharacteristic
292C3×C4#91111yescharacteristic
293C3×C4#91411yescharacteristic
294C3×C4#91811yescharacteristic
295C3×C4#91811yescharacteristic
296Dic3—41no—
297Dic3—41no—
298Dic3—122no—
299Dic3—121no—
300Dic3—121no—
301C32#159211yescharacteristic, complemented, Sylow
302C23—181no—
303Q8—61no—
304Q8—61no—
305D4—181no—
306D4—181no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—181no—
310C2×C4—361no—
311C2×C4—361no—
312C2×C4#379611yescharacteristic
313C2×C4C4×D3211yescharacteristic
314C2×C4#381111yescharacteristic
315C8—32no—
316C8—41no—
317C8—61no—
318C8—121no—
319C8—121no—
320C8—181no—
321C8—181no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3#472911yescharacteristic
326C2×C3#481711yescharacteristic
327C2×C3#482311yescharacteristic
328C2×C3#483811yescharacteristic
329C2×C3#488911yescharacteristic
330C2×C3#489411yescharacteristic
331D3—242no—
332D3—241no—
333D3—241no—
334C22—361no—
335C22—361no—
336C22—361no—
337C22#6388111yescharacteristic
338C4—21no—
339C4—121no—
340C4—121no—
341C4—361no—
342C4D3×ΓL(1,𝔽25)11yescharacteristic
343C4#6380411yescharacteristic
344C4#6393711yescharacteristic
345C4#6394511yescharacteristic
346C3—21no—
347C3#6822611yescharacteristic, complemented
348C3#6856511yescharacteristic, complemented
349C2—721no—
350C2#1059048411yescharacteristic
351C2#1059073211yescharacteristic
352C2#1059097811yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32C4×D4
prime radicalC2×C4⋊(C3⋊D3)C2×C4
cosocleC32×C42C23
hypercenterC4⋊C8D32
hypocenterC32#1592
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #10590259C2semidirect
4. C3×#4484C2semidirect
5. C3×#4479C2semidirect
7. #10589078C2semidirect
34. #4484D3semidirect
36. #4479D3semidirect
113. C3×C3⋊C8M16semidirect
244. C3⋊C8ΓL(1,𝔽25)semidirect
301. C32#1592semidirect
347. C3#68226semidirect
348. C3#68565semidirect

Maximal subgroups

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

subgroupindex
#105925372
C4×#37172
#105902592
C3×#44842
C3×#44792
#105890822
#105890782
#685653
#682263