Finite Group Databasebeta

← #11781114

Subgroups — #11781114

Order 1152.

404 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 88 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#10592782C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3#10590379C211yesmaximal, characteristic
4#10590327C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589092C211yesmaximal, characteristic, complemented
8#71371—31nomaximal
9#68443—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×#3755—21no—
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3717—21no—
15C2×#3710—21no—
16#63662—21no—
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#807—21no—
19C3×#807—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23#63566—21no—
24#63557—21no—
25C3×#500—31no—
26#5083—31no—
27#5083—31no—
28#5083—31no—
29#5083—61no—
30#5078D311yescharacteristic, complemented
31#4841—31no—
32#4834—31no—
33C8×D12—31no—
34C3×#332—31no—
35#4547—31no—
36#4547D311yescharacteristic, complemented
37#4486—31no—
38#4479—31no—
39#4458—31no—
40C2×C32×Q8—21no—
41C2×C4⋊(C3⋊D3)D411yescharacteristic
42C2×C4×C3⋊D3—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47#3755—42no—
48#3752—21no—
49C4⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3⋊(C3⋊C8)—21no—
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#3717—42no—
58#3710—42no—
59#1673—91noSylow
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—31no—
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4⋊Q8C2×D311yescharacteristic
64C2×#215—61no—
65C2×#215—61no—
66C2×#215—61no—
67C2×#215—62no—
68C2×#204—61no—
69C2×D3×C8—61no—
70C2×D3×C8—61no—
71C4×D12—31no—
72C4×D12—31no—
73C4×D12—61no—
74#869C2×D311yescharacteristic
75C3×#107—61no—
76C3×#107—61no—
77C3×C4⋊C8—31no—
78C3×C4⋊C8—31no—
79C3×C4×C8—31no—
80#835—61no—
81#835—61no—
82#835—61no—
83#835—62no—
84(C2×D3)⋊C8—61no—
85(C2×D3)⋊C8—61no—
86#816—61no—
87#807—21no—
88#807—21no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—31no—
91C4⋊(C3⋊C8)—61no—
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4⋊(C3⋊C8)C2×D311yescharacteristic
94C4×C3⋊C8C2×D311yescharacteristic
95C22×C3⋊D3—21no—
96C32×Q8—41no—
97C32×Q8—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4#4111yescharacteristic
102C2×C3⋊Dic3—21no—
103C4⋊(C3⋊D3)—21no—
104C4⋊(C3⋊D3)D812yes—
105C4×C3⋊D3—41no—
106C4×C3⋊D3—41no—
107C2×C3×Dic3—41no—
108C3×Dic6—41no—
109C3×Dic6—41no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8SD1612yescomplemented
115#500—31no—
116#461—91no—
117#461—91no—
118C8×D4—91no—
119#335—91no—
120#332—31no—
121#332—91no—
122C2×C3×Q8—22no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—61no—
126C2×D12—31no—
127C2×D12—31no—
128C2×D12—61no—
129C2×C4×D3—61no—
130C2×C4×D3—61no—
131C2×C4×D3—121no—
132C2×Dic6—21no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—61no—
136C2×C3×C8—61no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4—61no—
145C3×C42—21no—
146C3×C42C22×D311yescharacteristic
147C3×C42C22×D311yescharacteristic
148#215—122no—
149#215—122no—
150#215—122no—
151#215—124no—
152(C2×D3)⋊C4—61no—
153(C2×D3)⋊C4—61no—
154(C2×D3)⋊C4—121no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—61no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—62no—
163C2×C3⋊C8—61no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8#6711yescharacteristic
166C2×C3⋊C8#6711yescharacteristic
167#204—122no—
168D3×C8—122no—
169D3×C8—122no—
170C22×C32#12111yescharacteristic
171C2×C3⋊D3—22no—
172C2×C3⋊D3—41no—
173C2×C3⋊D3—41no—
174C32×C4—21no—
175C32×C4—81no—
176C32×C4C2×D811yescharacteristic
177C32×C4C2×SD1611yescharacteristic
178C32×C4#13511yescharacteristic
179C32×C4#13711yescharacteristic
180C3⋊Dic3—41no—
181C3×Dic3—81no—
182C2×SD16—181no—
183C2×SD16—181no—
184C22×C8—181no—
185C4⋊Q8—31no—
186C4⋊Q8D3211yescharacteristic
187C4×D4—91no—
188#107—61no—
189#107—61no—
190C4⋊C8—31no—
191C4⋊C8—31no—
192C4⋊C8—91no—
193#103—181no—
194#103—181no—
195C22⋊C8—181no—
196C4×C8—31no—
197C22×D3—61no—
198C22×D3—61no—
199C22×D3—121no—
200C3×Q8—42no—
201C3×Q8—42no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—41no—
205C3×Q8—41no—
206C3×Q8—121no—
207C3×Q8—121no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—121no—
215C2×C3×C4#23511yescharacteristic
216C2×C3×C4D3×D411yescharacteristic
217C2×C3×C4D3×D411yescharacteristic
218C2×C3×C4D3×D411yescharacteristic
219C2×C3×C4#23711yescharacteristic
220C2×C3×C4C2×#6711yescharacteristic
221C2×Dic3—41no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—121no—
225D12—32no—
226D12—32no—
227D12—62no—
228D12—61no—
229D12—61no—
230D12—121no—
231C4×D3—121no—
232C4×D3—121no—
233C4×D3—121no—
234C4×D3—121no—
235C4×D3—241no—
236C4×D3—241no—
237Dic6—41no—
238Dic6—41no—
239C3×C8—32no—
240C3×C8—61no—
241C3×C8—121no—
242C3×C8—121no—
243C3⋊C8—21no—
244C3⋊C8—41no—
245C3⋊C8—41no—
246C3⋊C8—122no—
247C3⋊C8—121no—
248C3⋊C8—121no—
249C3⋊C8#21512yescomplemented
250C2×C32#46111yescharacteristic
251C2×C32#46611yescharacteristic
252C2×C32#47311yescharacteristic
253C3⋊D3—42no—
254C2×Q8—21no—
255C2×Q8—61no—
256C2×D4—91no—
257C22×C4—181no—
258SD16—362no—
259SD16—362no—
260C2×C8—31no—
261C2×C8—31no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—181no—
265C2×C8—362no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—91no—
271C22⋊C4—181no—
272C42C2×D3211yescharacteristic
273C22×C3—21no—
274C22×C3#88411yescharacteristic
275C22×C3#93811yescharacteristic
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—122no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—241no—
284C2×D3—241no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—41no—
292C3×C4—82no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—241no—
296C3×C4D3×D811yescharacteristic
297C3×C4D3×D811yescharacteristic
298C3×C4#91211yescharacteristic
299C3×C4D3×SD1611yescharacteristic
300C3×C4#91611yescharacteristic
301C3×C4#91811yescharacteristic
302C3×C4C2×#21511yescharacteristic
303C3×C4#94211yescharacteristic
304Dic3—81no—
305Dic3—121no—
306Dic3—121no—
307Dic3—241no—
308C32#167311yescharacteristic, complemented, Sylow
309C23—181no—
310Q8—41no—
311Q8—41no—
312Q8—121no—
313Q8—121no—
314D4—92no—
315D4—181no—
316C2×C4—41no—
317C2×C4—121no—
318C2×C4—181no—
319C2×C4—361no—
320C2×C4—361no—
321C2×C4#380411yescharacteristic
322C2×C4D3×#6711yescharacteristic
323C2×C4#381111yescharacteristic
324C8—32no—
325C8—61no—
326C8—121no—
327C8—121no—
328C8—361no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#475511yescharacteristic
333C2×C3#477511yescharacteristic
334C2×C3#481811yescharacteristic
335C2×C3#508311yescharacteristic
336C2×C3#515711yescharacteristic
337C2×C3#519211yescharacteristic
338D3—122no—
339D3—122no—
340D3—242no—
341C22—182no—
342C22—361no—
343C22—361no—
344C22#6395811yescharacteristic
345C4—21no—
346C4—81no—
347C4—241no—
348C4—361no—
349C4#6392611yescharacteristic
350C4#6393611yescharacteristic
351C4D3×#21511yescharacteristic
352C4#6394811yescharacteristic
353C3—21no—
354C3#6844311yescharacteristic, complemented
355C3#7137111yescharacteristic, complemented
356C2—362no—
357C2#1059145511yescharacteristic
358C2#1059154311yescharacteristic
359C2#1059156511yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1673
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
2. C3×#5078C2semidirect
5. C3×#4547C2semidirect
6. #10589099C2semidirect
7. #10589092C2semidirect
30. #5078D3semidirect
36. #4547D3semidirect
114. C3×C3⋊C8SD16semidirect
249. C3⋊C8#215semidirect
308. C32#1673semidirect
354. C3#68443semidirect
355. C3#71371semidirect

Maximal subgroups

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

subgroupindex
#105927822
C3×#50782
#105903792
#105903272
C3×#45472
#105890992
#105890922
#713713
#684433