Finite Group Databasebeta

← #11777898

Subgroups — #11777898

Order 1152.

365 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C4×#3717C211yesmaximal, characteristic
2#10590338C211yesmaximal, characteristic
3#10590258C211yesmaximal, characteristic, complemented
4#10589325C211yesmaximal, characteristic
5C3×#4456C211yesmaximal, characteristic, complemented
6C3×#4454C211yesmaximal, characteristic, complemented
7#10589099C211yesmaximal, characteristic, complemented
8#68226—31nomaximal
9#68082—31nomaximal
10C4×C4⋊(C3⋊D3)C2211yescharacteristic
11C3×C4×Dic6C2211yescharacteristic
12C2×#3717C411yescharacteristic
13C2×#3712—21no—
14C32×C4×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊C8)C2211yescharacteristic
17C3×C4×C3⋊C8C2211yescharacteristic
18C3×C4×C3⋊C8C2211yescharacteristic
19#63573C411yescharacteristic
20Dic3⋊(C3⋊C8)—21no—
21#63566C411yescharacteristic
22#63564C411yescharacteristic
23#63557—21no—
24#63554—21no—
25C4×#215—31no—
26#5024—31no—
27#4838—31no—
28C4×#204—31no—
29#4688—31no—
30#4679—31no—
31C3×Q8⋊C8—31no—
32C3×#322—31no—
33#4479—31no—
34#4459—31no—
35#4459—31no—
36#4459—61no—
37#4458—31no—
38#4456D311yescharacteristic, complemented
39#4454D311yescharacteristic, complemented
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)—21no—
48C3×C4⋊Dic3C2×C411yescharacteristic
49C3×Dic3⋊C4—21no—
50C3×C4×Dic3—21no—
51C2×C3×C3⋊C8—21no—
52C2×C3×C3⋊C8C2×C411yescharacteristic
53C2×C3×C3⋊C8C2×C411yescharacteristic
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3717—22no—
57#3712—42no—
58#1592—91noSylow
59C3×C4×Q8—31no—
60C2×#215—31no—
61C2×#204—31no—
62C2×ΓL(1,𝔽25)—61no—
63C2×ΓL(1,𝔽25)—61no—
64C4×D12—31no—
65C4×D12—31no—
66C4×D12—61no—
67C4×Dic6C2×D311yescharacteristic
68C3×C4⋊C8—31no—
69C3×C4×C8—21no—
70C3×C4×C8—31no—
71C3×C4×C8—31no—
72C3×C4×C8C2×D311yescharacteristic
73C3×C4×C8C2×D311yescharacteristic
74#835—31no—
75#821—31no—
76(C2×D3)⋊C8—61no—
77(C2×D3)⋊C8—61no—
78#817—31no—
79#816—31no—
80#815—61no—
81Dic3⋊C8—61no—
82#809—31no—
83#808—31no—
84C4⋊(C3⋊C8)—31no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—61no—
87C4⋊(C3⋊C8)—61no—
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89#803—61no—
90C4×C3⋊C8C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3⋊D3—21no—
93C2×C32×C4C22×C411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3⋊Dic3—21no—
97C4⋊(C3⋊D3)—21no—
98C4⋊(C3⋊D3)—21no—
99C4×C3⋊D3—41no—
100C4×C3⋊D3—41no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C3×Dic6—21no—
104C3×Dic6—21no—
105C32×C8—41no—
106C3⋊(C3⋊C8)—41no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8M1612yescomplemented
112#446—91no—
113C4×SD16—91no—
114#436—91no—
115#335—91no—
116Q8⋊C8—31no—
117D4⋊C8—91no—
118#322—31no—
119C2×C3×Q8—31no—
120C2×D12—31no—
121C2×D12—31no—
122C2×D12—61no—
123C2×C4×D3—61no—
124C2×C4×D3—61no—
125C2×C4×D3—121no—
126C2×Dic6C4×D311yescharacteristic
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—31no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—41no—
134C2×C3×C8—61no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—61no—
137C3×C42—21no—
138C3×C42—61no—
139C3×C42C22×D311yescharacteristic
140C3×C42C22×D311yescharacteristic
141#215—62no—
142(C2×D3)⋊C4—61no—
143(C2×D3)⋊C4—61no—
144(C2×D3)⋊C4—121no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—61no—
148C4⋊Dic3C4×D311yescharacteristic
149Dic3⋊C4—21no—
150C4×Dic3—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—121no—
155C2×C3⋊C8C4×D311yescharacteristic
156C2×C3⋊C8C4×D311yescharacteristic
157C2×C3⋊C8D1211yescharacteristic
158C2×C3⋊C8D1211yescharacteristic
159#204—62no—
160ΓL(1,𝔽25)—122no—
161ΓL(1,𝔽25)—122no—
162C22×C32C4×D411yescharacteristic
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C32×C4—21no—
167C32×C4C2×M1611yescharacteristic
168C32×C4#13111yescharacteristic
169C32×C4AGL(1,ℤ/8)11yescharacteristic
170C32×C4#13711yescharacteristic
171C3⋊Dic3—41no—
172C3×Dic3—41no—
173C3×Dic3—41no—
174C2×SD16—91no—
175C2×M16—181no—
176C4×Q8—31no—
177C4×D4—91no—
178#106—91no—
179C4⋊C8—31no—
180C4⋊C8—91no—
181C4⋊C8—181no—
182#103—91no—
183#102—91no—
184C22⋊C8—181no—
185#97—181no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8D3211yescharacteristic
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—121no—
192C3×Q8—61no—
193C3×Q8—61no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4C2×C4×D311yescharacteristic
200C2×C3×C4C2×C4×D311yescharacteristic
201C2×C3×C4C2×D1211yescharacteristic
202C2×C3×C4#23511yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4#23711yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—121no—
210D12—62no—
211D12—61no—
212D12—61no—
213D12—61no—
214D12—61no—
215D12—121no—
216C4×D3—121no—
217C4×D3—121no—
218C4×D3—121no—
219C4×D3—121no—
220C4×D3—241no—
221C4×D3—241no—
222Dic6—21no—
223Dic6—21no—
224C3×C8—32no—
225C3×C8—41no—
226C3×C8—41no—
227C3×C8—42no—
228C3×C8—61no—
229C3×C8—61no—
230C3×C8—61no—
231C3×C8—121no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—41no—
236C3⋊C8—121no—
237C3⋊C8—121no—
238C3⋊C8—241no—
239C3⋊C8ΓL(1,𝔽25)12yescomplemented
240C2×C32#43611yescharacteristic
241C2×C32#44011yescharacteristic
242C2×C32#44411yescharacteristic
243C3⋊D3—81no—
244C2×Q8—31no—
245C2×D4—91no—
246C22×C4—181no—
247SD16—182no—
248M16—362no—
249C2×C8—21no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—61no—
255C2×C8—181no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—91no—
259C22⋊C4—181no—
260C42—61no—
261C42C2×D3211yescharacteristic
262C22×C3—21no—
263C22×C3C4×D1211yescharacteristic
264C22×C3#67⋊C411yescharacteristic
265C2×D3—122no—
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—121no—
269C2×D3—121no—
270C2×D3—121no—
271C2×D3—121no—
272C2×D3—241no—
273C2×D3—241no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—121no—
282C3×C4—121no—
283C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
284C3×C4#90111yescharacteristic
285C3×C4#90111yescharacteristic
286C3×C4D3×M1611yescharacteristic
287C3×C4#90811yescharacteristic
288C3×C4#90911yescharacteristic
289C3×C4#91411yescharacteristic
290C3×C4#91811yescharacteristic
291Dic3—41no—
292Dic3—41no—
293Dic3—121no—
294Dic3—121no—
295Dic3—241no—
296C32#159211yescharacteristic, complemented, Sylow
297C23—181no—
298Q8—61no—
299Q8—61no—
300D4—181no—
301D4—181no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—181no—
305C2×C4—361no—
306C2×C4—361no—
307C2×C4C4×D3211yescharacteristic
308C2×C4D3×D1211yescharacteristic
309C2×C4#380411yescharacteristic
310C8—32no—
311C8—41no—
312C8—61no—
313C8—61no—
314C8—61no—
315C8—121no—
316C8—361no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3#468811yescharacteristic
321C2×C3#471011yescharacteristic
322C2×C3#471711yescharacteristic
323C2×C3#472911yescharacteristic
324C2×C3#481711yescharacteristic
325C2×C3#482311yescharacteristic
326D3—242no—
327D3—241no—
328D3—241no—
329C22—361no—
330C22—361no—
331C22—361no—
332C22#6388011yescharacteristic
333C4—21no—
334C4—121no—
335C4—121no—
336C4—361no—
337C4D3×ΓL(1,𝔽25)11yescharacteristic
338C4#6379411yescharacteristic
339C4#6380111yescharacteristic
340C4#6380311yescharacteristic
341C3—21no—
342C3#6808211yescharacteristic, complemented
343C3#6822611yescharacteristic, complemented
344C2—721no—
345C2#1059034211yescharacteristic
346C2#1059073011yescharacteristic
347C2#1059079511yescharacteristic
subgroupG/H
centerC2×C4D3×D12
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. #10590258C2semidirect
5. C3×#4456C2semidirect
6. C3×#4454C2semidirect
7. #10589099C2semidirect
38. #4456D3semidirect
39. #4454D3semidirect
111. C3×C3⋊C8M16semidirect
239. C3⋊C8ΓL(1,𝔽25)semidirect
296. C32#1592semidirect
342. C3#68082semidirect
343. C3#68226semidirect

Maximal subgroups

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

subgroupindex
C4×#37172
#105903382
#105902582
#105893252
C3×#44562
C3×#44542
#105890992
#682263
#680823