Finite Group Databasebeta

← #11781933

Subgroups — #11781933

Order 1152.

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

normal subgroups only

This group has 364 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
1C3×#5083C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3#10590363C211yesmaximal, characteristic
4#10590337C211yesmaximal, characteristic
5#10589413C211yesmaximal, characteristic, complemented
6#10589097C211yesmaximal, characteristic, complemented
7#10589093C211yesmaximal, characteristic, complemented
8#71373—31nomaximal
9#71041—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×C3×#215—21no—
12C3×C4×D12C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3715—21no—
15C2×#3711—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#835—21no—
19C3×#807—21no—
20C3×#807—21no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23#63565—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#500—31no—
26C3×#461—31no—
27#5083—31no—
28#5083D311yescharacteristic
29#5078D311yescharacteristic, complemented
30#5019—31no—
31#5012—31no—
32#4688—31no—
33#4547—31no—
34#4547—31no—
35#4547—31no—
36#4547—61no—
37#4482—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)—21no—
48C2×C3⋊(C3⋊C8)—21no—
49C3×#215—42no—
50C3×(C2×D3)⋊C4—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3D411yescharacteristic
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57#3715—42no—
58#3711—42no—
59#1693—91noSylow
60C2×C3×SD16—61no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—31no—
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4⋊Q8C2×D311yescharacteristic
65C3×C4×D4—31no—
66C2×#215—21no—
67C2×#215—61no—
68C2×#214—61no—
69C2×#208—61no—
70C2×ΓL(1,𝔽25)—61no—
71C4×D12C2×D311yescharacteristic
72#869C2×D311yescharacteristic
73C3×#107—61no—
74C3×#107—61no—
75C3×C4⋊C8—31no—
76C3×#103—61no—
77C3×C4×C8—31no—
78#835—21no—
79#835—61no—
80#830—61no—
81(C2×D3)⋊C8—61no—
82#810—61no—
83#807—21no—
84#807—21no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3×D3—21no—
94C32×Q8—41no—
95C32×Q8—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C2×C3×Dic3—21no—
101C2×C3×Dic3—41no—
102C3×D12—21no—
103C3×D12—21no—
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108C3⋊(C3⋊C8)—41no—
109C3⋊(C3⋊C8)—41no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8SD1612yescomplemented
113#500—31no—
114#461—31no—
115#461—91no—
116#436—91no—
117#334—91no—
118#332—91no—
119#332—91no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—41no—
123C2×C3×Q8—61no—
124C2×C3×D4—31no—
125C22×C3×C4—61no—
126C2×D12#6711yescharacteristic
127C2×C4×D3—21no—
128C2×Dic6—21no—
129C3×SD16—122no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—61no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—61no—
142C3×C22⋊C4—61no—
143C3×C42—21no—
144C3×C42C22×D311yescharacteristic
145C3×C42C22×D311yescharacteristic
146#215—42no—
147#215—122no—
148#214—122no—
149(C2×D3)⋊C4—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3#6711yescharacteristic
153#208—122no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—121no—
160C2×C3⋊C8—121no—
161C2×C3⋊C8#6711yescharacteristic
162C2×C3⋊C8#6711yescharacteristic
163ΓL(1,𝔽25)—122no—
164C22×C32#12111yescharacteristic
165C2×C3×D3—41no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C32×C4—21no—
169C32×C4—81no—
170C32×C4C2×SD1611yescharacteristic
171C32×C4AGL(1,ℤ/8)11yescharacteristic
172C32×C4AGL(1,ℤ/8)11yescharacteristic
173C32×C4#13711yescharacteristic
174C3×Dic3—41no—
175C3×Dic3—81no—
176C2×SD16—61no—
177C2×SD16—181no—
178C2×M16—181no—
179C4⋊Q8—31no—
180C4⋊Q8D3211yescharacteristic
181C4×D4—31no—
182#107—61no—
183#107—61no—
184C4⋊C8—31no—
185C4⋊C8—91no—
186C4⋊C8—91no—
187#103—61no—
188#103—181no—
189C22⋊C8—181no—
190C4×C8—31no—
191C23×C3—61no—
192C22×D3—21no—
193C3×Q8—41no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—81no—
198C3×Q8—81no—
199C3×Q8—121no—
200C3×Q8—121no—
201C3×D4—61no—
202C3×D4—61no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—61no—
210C2×C3×C4—121no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4D3×D411yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4#23711yescharacteristic
216C2×C3×C4#23711yescharacteristic
217C2×C3×C4C2×#6711yescharacteristic
218C2×C3×C4C2×#6711yescharacteristic
219C2×Dic3—21no—
220C2×Dic3—41no—
221D12—21no—
222D12—21no—
223C4×D3—41no—
224C4×D3—41no—
225Dic6—41no—
226Dic6—41no—
227C3×C8—32no—
228C3×C8—61no—
229C3×C8—121no—
230C3⋊C8—21no—
231C3⋊C8—41no—
232C3⋊C8—121no—
233C3⋊C8—121no—
234C3⋊C8—121no—
235C3⋊C8—121no—
236C3⋊C8—241no—
237C3⋊C8—241no—
238C3⋊C8#21512yescomplemented
239C2×C32#46111yescharacteristic
240C2×C32#46911yescharacteristic
241C2×C32#47311yescharacteristic
242C3×D3—81no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×D4—31no—
246C22×C4—61no—
247SD16—122no—
248SD16—362no—
249M16—362no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—61no—
253C2×C8—181no—
254C2×C8—181no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C22⋊C4—61no—
261C42C2×D3211yescharacteristic
262C22×C3—21no—
263C22×C3—121no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3#93811yescharacteristic
267C22×C3#93911yescharacteristic
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—82no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—121no—
282C3×C4—241no—
283C3×C4#91111yescharacteristic
284C3×C4#91111yescharacteristic
285C3×C4D3×SD1611yescharacteristic
286C3×C4#91511yescharacteristic
287C3×C4#93211yescharacteristic
288C3×C4C2×#21511yescharacteristic
289C3×C4#94211yescharacteristic
290C3×C4#94911yescharacteristic
291Dic3—41no—
292Dic3—81no—
293C32#169311yescharacteristic, complemented, Sylow
294C23—61no—
295Q8—41no—
296Q8—41no—
297Q8—121no—
298Q8—121no—
299D4—61no—
300D4—61no—
301C2×C4—41no—
302C2×C4—61no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4#380511yescharacteristic
307C2×C4D3×#6711yescharacteristic
308C2×C4D3×#6711yescharacteristic
309C8—32no—
310C8—61no—
311C8—121no—
312C8—361no—
313C8—361no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—241no—
318C2×C3#503911yescharacteristic
319C2×C3#508311yescharacteristic
320C2×C3#515911yescharacteristic
321C2×C3#516211yescharacteristic
322C2×C3#519211yescharacteristic
323C2×C3#520411yescharacteristic
324D3—81no—
325C22—121no—
326C22—121no—
327C22—121no—
328C22#6397011yescharacteristic
329C4—21no—
330C4—81no—
331C4—121no—
332C4—241no—
333C4#6392511yescharacteristic
334C4#6393211yescharacteristic
335C4D3×#21511yescharacteristic
336C4#6394611yescharacteristic
337C3—21no—
338C3#7104111yescharacteristic, complemented
339C3#7137311yescharacteristic, complemented
340C2—241no—
341C2#1059145111yescharacteristic
342C2#1059156411yescharacteristic
343C2#1059164111yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1693
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. #10589413C2semidirect
6. #10589097C2semidirect
7. #10589093C2semidirect
29. #5078D3semidirect
112. C3×C3⋊C8SD16semidirect
238. C3⋊C8#215semidirect
293. C32#1693semidirect
338. C3#71041semidirect
339. C3#71373semidirect

Maximal subgroups

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

subgroupindex
C3×#50832
C3×#50782
#105903632
#105903372
#105894132
#105890972
#105890932
#713733
#710413