Finite Group Databasebeta

← #11781353

Subgroups — #11781353

Order 1152.

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

normal subgroups only

This group has 392 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#10592782C211yesmaximal, characteristic
2C3×#5093C211yesmaximal, characteristic, complemented
3#10590327C211yesmaximal, characteristic
4#10590313C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589097C211yesmaximal, characteristic, complemented
7#10589093C211yesmaximal, characteristic, complemented
8#71232—31nomaximal
9#68445—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×#3755—21no—
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3710—21no—
15#63662—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#5083—31no—
29#5083—31no—
30#5083—61no—
31#4846—31no—
32#4834—31no—
33C8×D12—31no—
34C3×#332—31no—
35#4547—31no—
36#4547D311yescharacteristic, complemented
37#4482—31no—
38#4480—31no—
39#4457—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#3710—42no—
58#1679—91noSylow
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C2×#215—61no—
64C2×#215—61no—
65C2×#215—62no—
66C2×D3×C8—61no—
67C2×D3×C8—61no—
68C4×D12—31no—
69C4×D12—31no—
70C4×D12—61no—
71#869C2×D311yescharacteristic
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×#103—61no—
75C3×#103—61no—
76C3×C4×C8—31no—
77#835—61no—
78#835—61no—
79#835—62no—
80#821—61no—
81(C2×D3)⋊C8—61no—
82(C2×D3)⋊C8—61no—
83#818—61no—
84#810—21no—
85#810—21no—
86#809—61no—
87#807—61no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4×C3⋊C8C2×D311yescharacteristic
94C22×C3⋊D3—21no—
95C32×Q8—41no—
96C32×Q8—41no—
97C2×C32×C4—41no—
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4#4111yescharacteristic
101C2×C3⋊Dic3—21no—
102C4⋊(C3⋊D3)—21no—
103C4⋊(C3⋊D3)SD1612yes—
104C4×C3⋊D3—41no—
105C4×C3⋊D3—41no—
106C2×C3×Dic3—41no—
107C3×Dic6—41no—
108C3×Dic6—41no—
109C3⋊(C3⋊C8)—41no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8—41no—
114#487—31no—
115#474—91no—
116#461—91no—
117C8×D4—91no—
118#334—91no—
119#332—31no—
120#332—91no—
121C2×C3×Q8—22no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—61no—
125C2×D12—31no—
126C2×D12—31no—
127C2×D12—61no—
128C2×C4×D3—61no—
129C2×C4×D3—61no—
130C2×C4×D3—121no—
131C2×Dic6—21no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—61no—
135C2×C3×C8—61no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—61no—
143C3×C4⋊C4—61no—
144C3×C42—21no—
145C3×C42C22×D311yescharacteristic
146C3×C42C22×D311yescharacteristic
147#215—122no—
148#215—122no—
149#215—124no—
150(C2×D3)⋊C4—61no—
151(C2×D3)⋊C4—61no—
152(C2×D3)⋊C4—121no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—61no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—62no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—61no—
163C2×C3⋊C8#6711yescharacteristic
164C2×C3⋊C8#6711yescharacteristic
165D3×C8—122no—
166D3×C8—122no—
167C22×C32#12111yescharacteristic
168C2×C3⋊D3—22no—
169C2×C3⋊D3—41no—
170C2×C3⋊D3—41no—
171C32×C4—21no—
172C32×C4—81no—
173C32×C4C2×SD1611yescharacteristic
174C32×C4#13511yescharacteristic
175C32×C4#13511yescharacteristic
176C32×C4AGL(1,ℤ/8)11yescharacteristic
177C3⋊Dic3—41no—
178C3×Dic3—81no—
179C2×SD16—181no—
180C22×C8—181no—
181C4⋊Q8—31no—
182C4⋊Q8D3211yescharacteristic
183C4×D4—91no—
184#107—181no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—91no—
188#103—61no—
189#103—61no—
190#103—181no—
191#102—181no—
192C22⋊C8—181no—
193C4×C8—31no—
194C22×D3—61no—
195C22×D3—61no—
196C22×D3—121no—
197C3×Q8—42no—
198C3×Q8—42no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—41no—
203C3×Q8—121no—
204C3×Q8—121no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—42no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—121no—
212C2×C3×C4#23511yescharacteristic
213C2×C3×C4D3×D411yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4D3×D411yescharacteristic
216C2×C3×C4#23711yescharacteristic
217C2×C3×C4C2×#6711yescharacteristic
218C2×Dic3—41no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—121no—
222D12—32no—
223D12—32no—
224D12—62no—
225D12—61no—
226D12—61no—
227D12—121no—
228C4×D3—121no—
229C4×D3—121no—
230C4×D3—121no—
231C4×D3—121no—
232C4×D3—241no—
233C4×D3—241no—
234Dic6—41no—
235Dic6—41no—
236C3×C8—61no—
237C3×C8—61no—
238C3×C8—121no—
239C3×C8—121no—
240C3⋊C8—21no—
241C3⋊C8—21no—
242C3⋊C8—41no—
243C3⋊C8—41no—
244C3⋊C8—122no—
245C3⋊C8—121no—
246C3⋊C8—121no—
247C2×C32#46311yescharacteristic
248C2×C32#46511yescharacteristic
249C2×C32#47311yescharacteristic
250C3⋊D3—42no—
251C2×Q8—21no—
252C2×Q8—61no—
253C2×D4—91no—
254C22×C4—181no—
255SD16—362no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—181no—
261C2×C8—362no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—91no—
267C22⋊C4—181no—
268C42C2×D3211yescharacteristic
269C22×C3—21no—
270C22×C3#88411yescharacteristic
271C22×C3#93811yescharacteristic
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—122no—
275C2×D3—121no—
276C2×D3—121no—
277C2×D3—121no—
278C2×D3—121no—
279C2×D3—241no—
280C2×D3—241no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—82no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4—241no—
292C3×C4#91211yescharacteristic
293C3×C4D3×SD1611yescharacteristic
294C3×C4D3×SD1611yescharacteristic
295C3×C4#91411yescharacteristic
296C3×C4#91611yescharacteristic
297C3×C4#91911yescharacteristic
298C3×C4#94911yescharacteristic
299C3×C4#95011yescharacteristic
300Dic3—81no—
301Dic3—121no—
302Dic3—121no—
303Dic3—241no—
304C32#167911yescharacteristic, complemented, Sylow
305C23—181no—
306Q8—41no—
307Q8—41no—
308Q8—121no—
309Q8—121no—
310D4—92no—
311D4—181no—
312C2×C4—41no—
313C2×C4—121no—
314C2×C4—181no—
315C2×C4—361no—
316C2×C4—361no—
317C2×C4#380411yescharacteristic
318C2×C4D3×#6711yescharacteristic
319C2×C4#381111yescharacteristic
320C8—61no—
321C8—61no—
322C8—121no—
323C8—121no—
324C8—361no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#475511yescharacteristic
329C2×C3#477811yescharacteristic
330C2×C3#481911yescharacteristic
331C2×C3#505711yescharacteristic
332C2×C3#517111yescharacteristic
333C2×C3#519211yescharacteristic
334D3—122no—
335D3—122no—
336D3—242no—
337C22—182no—
338C22—361no—
339C22—361no—
340C22#6395811yescharacteristic
341C4—21no—
342C4—81no—
343C4—241no—
344C4—361no—
345C4#6393011yescharacteristic
346C4#6393611yescharacteristic
347C4#6393911yescharacteristic
348C4#6394911yescharacteristic
349C3—21no—
350C3#6844511yescharacteristic, complemented
351C3#7123211yescharacteristic, complemented
352C2—362no—
353C2#1059146511yescharacteristic
354C2#1059154311yescharacteristic
355C2#1059156811yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1679
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×#4547C2semidirect
6. #10589097C2semidirect
7. #10589093C2semidirect
26. #5093D3semidirect
36. #4547D3semidirect
304. C32#1679semidirect
350. C3#68445semidirect
351. C3#71232semidirect

Maximal subgroups

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

subgroupindex
#105927822
C3×#50932
#105903272
#105903132
C3×#45472
#105890972
#105890932
#712323
#684453