Finite Group Databasebeta

← #11781539

Subgroups — #11781539

Order 1152.

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

normal subgroups only

This group has 374 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#10592783C211yesmaximal, characteristic
2C3×#5093C211yesmaximal, characteristic, complemented
3#10590327C211yesmaximal, characteristic
4#10590313C211yesmaximal, characteristic
5C3×#4546C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589093C211yesmaximal, characteristic, complemented
8#71233—31nomaximal
9#68499—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C4×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3710—21no—
14#63636—21no—
15#63635—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#810—21no—
18C3×#810—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63575—21no—
23#63564—21no—
24#63557—21no—
25C3×#487—31no—
26#5093D311yescharacteristic, complemented
27#5087—31no—
28#5084—31no—
29#5084—31no—
30#5084—61no—
31#4846—31no—
32#4834—31no—
33C8×D12—31no—
34C3×#333—31no—
35#4547—31no—
36#4546D311yescharacteristic, complemented
37#4482—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#3752—21no—
48C4⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊C8)—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3710—42no—
57#1683—91noSylow
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C2×D311yescharacteristic
62C2×D3×C8—61no—
63C2×D3×C8—61no—
64C4×D12—31no—
65C4×D12—31no—
66C4×D12—61no—
67#869C2×D311yescharacteristic
68C3×C4⋊C8—31no—
69C3×C4⋊C8—31no—
70C3×#103—61no—
71C3×#103—61no—
72C3×C4×C8—31no—
73#821—61no—
74(C2×D3)⋊C8—61no—
75(C2×D3)⋊C8—61no—
76#818—61no—
77#810—21no—
78#810—21no—
79#809—61no—
80#809—61no—
81#809—61no—
82#809—62no—
83#808—61no—
84#808—61no—
85#808—62no—
86#807—61no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3⋊D3—21no—
94C32×Q8—81no—
95C2×C32×C4—41no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C3⋊Dic3—21no—
100C4⋊(C3⋊D3)—21no—
101C4⋊(C3⋊D3)Q1612yes—
102C4×C3⋊D3—41no—
103C4×C3⋊D3—41no—
104C2×C3×Dic3—41no—
105C3×Dic6—41no—
106C3×Dic6—41no—
107C3⋊(C3⋊C8)—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8—41no—
112#487—31no—
113#476—91no—
114#474—91no—
115C8×D4—91no—
116#335—91no—
117#333—31no—
118#332—91no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—41no—
122C2×C3×Q8—61no—
123C2×D12—31no—
124C2×D12—31no—
125C2×D12—61no—
126C2×C4×D3—61no—
127C2×C4×D3—61no—
128C2×C4×D3—121no—
129C2×Dic6—21no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—61no—
133C2×C3×C8—61no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—61no—
142C3×C42—21no—
143C3×C42C22×D311yescharacteristic
144C3×C42C22×D311yescharacteristic
145(C2×D3)⋊C4—61no—
146(C2×D3)⋊C4—61no—
147(C2×D3)⋊C4—121no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152C4⋊Dic3—61no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—62no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8#6711yescharacteristic
159C2×C3⋊C8#6711yescharacteristic
160D3×C8—122no—
161D3×C8—122no—
162C22×C32#12111yescharacteristic
163C2×C3⋊D3—22no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C32×C4—21no—
167C32×C4—81no—
168C32×C4C2×Q1611yescharacteristic
169C32×C4#13511yescharacteristic
170C32×C4#13511yescharacteristic
171C32×C4AGL(1,ℤ/8)11yescharacteristic
172C3⋊Dic3—41no—
173C3×Dic3—81no—
174C22×C8—181no—
175C4⋊Q8—31no—
176C4⋊Q8D3211yescharacteristic
177C4×D4—91no—
178#107—181no—
179#106—181no—
180C4⋊C8—31no—
181C4⋊C8—31no—
182C4⋊C8—91no—
183#103—61no—
184#103—61no—
185#102—181no—
186#102—181no—
187C22⋊C8—181no—
188C4×C8—31no—
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—121no—
192C3×Q8—81no—
193C3×Q8—81no—
194C3×Q8—82no—
195C3×Q8—121no—
196C3×Q8—121no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—121no—
204C2×C3×C4#23511yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4#23711yescharacteristic
209C2×C3×C4C2×#6711yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—121no—
214D12—32no—
215D12—32no—
216D12—62no—
217D12—61no—
218D12—61no—
219D12—121no—
220C4×D3—121no—
221C4×D3—121no—
222C4×D3—121no—
223C4×D3—121no—
224C4×D3—241no—
225C4×D3—241no—
226Dic6—41no—
227Dic6—41no—
228C3×C8—61no—
229C3×C8—61no—
230C3×C8—121no—
231C3×C8—121no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—41no—
235C3⋊C8—41no—
236C3⋊C8—122no—
237C3⋊C8—121no—
238C3⋊C8—121no—
239C2×C32#46311yescharacteristic
240C2×C32#46711yescharacteristic
241C2×C32#47111yescharacteristic
242C3⋊D3—42no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×D4—91no—
246C22×C4—181no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—181no—
252C2×C8—362no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—91no—
258C22⋊C4—181no—
259C42C2×D3211yescharacteristic
260C22×C3—21no—
261C22×C3#88411yescharacteristic
262C22×C3#93811yescharacteristic
263C2×D3—62no—
264C2×D3—62no—
265C2×D3—122no—
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—121no—
269C2×D3—121no—
270C2×D3—241no—
271C2×D3—241no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—82no—
280C3×C4—81no—
281C3×C4—81no—
282C3×C4—241no—
283C3×C4#91411yescharacteristic
284C3×C4#91611yescharacteristic
285C3×C4D3×Q1611yescharacteristic
286C3×C4D3×Q1611yescharacteristic
287C3×C4#91911yescharacteristic
288C3×C4#91911yescharacteristic
289C3×C4#94911yescharacteristic
290C3×C4#95011yescharacteristic
291Dic3—81no—
292Dic3—121no—
293Dic3—121no—
294Dic3—241no—
295C32#168311yescharacteristic, complemented, Sylow
296C23—181no—
297Q8—81no—
298Q8—121no—
299Q8—121no—
300D4—92no—
301D4—181no—
302C2×C4—41no—
303C2×C4—121no—
304C2×C4—181no—
305C2×C4—361no—
306C2×C4—361no—
307C2×C4#380411yescharacteristic
308C2×C4D3×#6711yescharacteristic
309C2×C4#381111yescharacteristic
310C8—61no—
311C8—61no—
312C8—121no—
313C8—121no—
314C8—361no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#474811yescharacteristic
319C2×C3#481311yescharacteristic
320C2×C3#481911yescharacteristic
321C2×C3#505711yescharacteristic
322C2×C3#517711yescharacteristic
323C2×C3#518811yescharacteristic
324D3—122no—
325D3—122no—
326D3—242no—
327C22—182no—
328C22—361no—
329C22—361no—
330C22#6395811yescharacteristic
331C4—21no—
332C4—81no—
333C4—241no—
334C4—361no—
335C4#6393911yescharacteristic
336C4#6394411yescharacteristic
337C4#6394911yescharacteristic
338C4#6395011yescharacteristic
339C3—21no—
340C3#6849911yescharacteristic, complemented
341C3#7123311yescharacteristic, complemented
342C2—362no—
343C2#1059140711yescharacteristic
344C2#1059156811yescharacteristic
345C2#1059160111yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1683
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×#5093C2semidirect
5. C3×#4546C2semidirect
6. #10589099C2semidirect
7. #10589093C2semidirect
26. #5093D3semidirect
36. #4546D3semidirect
295. C32#1683semidirect
340. C3#68499semidirect
341. C3#71233semidirect

Maximal subgroups

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

subgroupindex
#105927832
C3×#50932
#105903272
#105903132
C3×#45462
#105890992
#105890932
#712333
#684993