Finite Group Databasebeta

← #11779959

Subgroups — #11779959

Order 1152.

465 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 90 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#10593960C211yesmaximal, characteristic
2#10592792C211yesmaximal, characteristic, complemented
3C3×#5090C211yesmaximal, characteristic, complemented
4C3×#5087C211yesmaximal, characteristic
5#10589096C211yesmaximal, characteristic, complemented
6#10589083C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic
8#71113—31nomaximal
9#71062—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#216—21no—
13C3×C4×D12C2211yescharacteristic
14C3×C4×Dic6C2211yescharacteristic
15#63896—22no—
16Dic3×D12—21no—
17D3×C4⋊Dic3—21no—
18#63869—21no—
19#63856—21no—
20#63637—22no—
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#835—21no—
23C3×#809—21no—
24C3×#807—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4⋊(C3⋊C8)C2211yescharacteristic
27#5579—31no—
28#5543—31no—
29C3×#474—31no—
30C3×#462—31no—
31#5093—31no—
32#5093—31no—
33#5093—61no—
34#5090D311yescharacteristic, complemented
35#5087D311yescharacteristic
36#4535—31no—
37D4⋊(C3⋊C8)—31no—
38#4484—31no—
39#4483—31no—
40#4481—31no—
41#4481—31no—
42C2×C32×Q8—21no—
43C2×#634D412yes—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×Dic6D411yescharacteristic
47C2×D3×Dic3—22no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C42C2311yescharacteristic
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)—21no—
53C2×C3⋊(C3⋊C8)—21no—
54C3×#216—42no—
55C3×(C2×D3)⋊C4—21no—
56C3×C4⋊Dic3D411yescharacteristic
57C3×C4⋊Dic3D411yescharacteristic
58C3×Dic3⋊C4—21no—
59C3×C4×Dic3—21no—
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8—21no—
62#3724—22no—
63Dic3⋊Dic3—22no—
64(C2×D3)⋊Dic3—22no—
65#1644—91noSylow
66C2×C3×Q16—61no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8C2×D311yescharacteristic
69C3×C4⋊Q8C2×D311yescharacteristic
70C3×C4×Q8—31no—
71C3×C4×D4—31no—
72#946—61no—
73C2×#216—21no—
74D4×Dic3—61no—
75C2×C4⋊Dic3—61no—
76#924—61no—
77#896—62no—
78D3×C4⋊C4—61no—
79#890—61no—
80#878—62no—
81C4×D12—61no—
82C4×D12C2×D311yescharacteristic
83#870—61no—
84#869—31no—
85#869—31no—
86#869—61no—
87C4×Dic6C2×D311yescharacteristic
88C3×#107—61no—
89C3×C4⋊C8—31no—
90C3×C4⋊C8—31no—
91C3×#103—61no—
92C3×#102—61no—
93#835—21no—
94#810—62no—
95#810—62no—
96#810—64no—
97#809—21no—
98#807—21no—
99C4⋊(C3⋊C8)C2×D311yescharacteristic
100C4⋊(C3⋊C8)C2×D311yescharacteristic
101C4×C3⋊C8—31no—
102C4×C3⋊C8—31no—
103C4×C3⋊C8—61no—
104C22×C3×D3—21no—
105C32×Q8—41no—
106C32×Q8—41no—
107C2×C32×C4—41no—
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C3⋊Dic3—22no—
112#634—42no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C3×D12—21no—
117C3×D12Q1612yescomplemented
118C3×C4×D3—41no—
119C3×C4×D3—41no—
120C3×Dic6—21no—
121C3×Dic6—21no—
122D3×Dic3—82no—
123C3⋊(C3⋊C8)—41no—
124C3×C3⋊C8—41no—
125C3×C3⋊C8—41no—
126#554—91no—
127#487—91no—
128#474—31no—
129#462—31no—
130#333—91no—
131Q8⋊C8—91no—
132D4⋊C8—91no—
133C2×C3×Q8—21no—
134C2×C3×Q8—21no—
135C2×C3×Q8—31no—
136C2×C3×Q8—41no—
137C2×C3×D4—31no—
138C22×C3×C4—61no—
139C22×Dic3—62no—
140C2×D12#6711yescharacteristic
141C2×C4×D3—21no—
142C2×C4×D3—62no—
143C2×Dic6—32no—
144C2×Dic6—32no—
145C2×Dic6—62no—
146C2×Dic6#6711yescharacteristic
147C3×Q16—122no—
148C2×C3×C8—61no—
149C2×C3×C8—61no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—61no—
159C3×C22⋊C4—61no—
160C3×C42—21no—
161C3×C42—61no—
162C3×C42C22×D311yescharacteristic
163C3×C42C22×D311yescharacteristic
164#217—62no—
165#216—42no—
166(C2×D3)⋊C4—21no—
167(C2×D3)⋊C4—62no—
168C4⋊Dic3—62no—
169C4⋊Dic3—61no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3—61no—
173C4⋊Dic3—121no—
174C4⋊Dic3—121no—
175C4⋊Dic3#6711yescharacteristic
176C4⋊Dic3#6711yescharacteristic
177Dic3⋊C4—21no—
178Dic3⋊C4—62no—
179Dic3⋊C4—62no—
180Dic3⋊C4—62no—
181C4×Dic3—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—62no—
185C2×C3⋊C8—61no—
186C2×C3⋊C8—61no—
187C22×C32#12011yescharacteristic
188C2×C3×D3—22no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C32×C4—21no—
192C32×C4—81no—
193C32×C4C2×Q1611yescharacteristic
194C32×C4#13511yescharacteristic
195C32×C4AGL(1,ℤ/8)11yescharacteristic
196C32×C4#13711yescharacteristic
197C3⋊Dic3—42no—
198C3×Dic3—41no—
199C3×Dic3—41no—
200C3×Dic3—41no—
201C2×Q16—61no—
202C4⋊Q8—91no—
203C4⋊Q8D3211yescharacteristic
204#125—181no—
205C22⋊Q8—182no—
206C22⋊Q8—181no—
207C4×Q8—31no—
208C4×D4—31no—
209C4×D4—181no—
210C2×C4⋊C4—181no—
211#107—61no—
212C4⋊C8—31no—
213C4⋊C8—31no—
214#103—61no—
215#103—182no—
216#102—61no—
217C4×C8—91no—
218C23×C3—61no—
219C22×D3—21no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×Q8—61no—
225C3×Q8—61no—
226C3×Q8—81no—
227C3×Q8—81no—
228C3×D4—32no—
229C3×D4—61no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—121no—
240C2×C3×C4—121no—
241C2×C3×C4D3×D411yescharacteristic
242C2×C3×C4D3×D411yescharacteristic
243C2×C3×C4D3×D411yescharacteristic
244C2×C3×C4D3×D411yescharacteristic
245C2×C3×C4C2×#6711yescharacteristic
246C2×C3×C4C2×#6711yescharacteristic
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—122no—
253C2×Dic3—242no—
254D12—21no—
255D12#21612yescomplemented
256C4×D3—41no—
257C4×D3—41no—
258C4×D3—242no—
259Dic6—21no—
260Dic6—21no—
261Dic6—122no—
262Dic6—122no—
263Dic6—124no—
264C3×C8—121no—
265C3×C8—121no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C3⋊C8—122no—
269C3⋊C8—121no—
270C3⋊C8—121no—
271C2×C32#44911yescharacteristic
272C2×C32#45111yescharacteristic
273C2×C32#45511yescharacteristic
274C3×D3—42no—
275C2×Q8—21no—
276C2×Q8—31no—
277C2×Q8—92no—
278C2×D4—31no—
279C22×C4—61no—
280C22×C4—182no—
281Q16—122no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—181no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—61no—
290C4⋊C4—182no—
291C4⋊C4—182no—
292C4⋊C4—181no—
293C4⋊C4—181no—
294C22⋊C4—61no—
295C22⋊C4—182no—
296C42—61no—
297C42C2×D3211yescharacteristic
298C22×C3—21no—
299C22×C3—62no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3#93711yescharacteristic
303C22×C3#93711yescharacteristic
304C2×D3—22no—
305C2×D3—41no—
306C2×D3—41no—
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—121no—
320C3×C4#91111yescharacteristic
321C3×C4#91211yescharacteristic
322C3×C4D3×Q1611yescharacteristic
323C3×C4#91811yescharacteristic
324C3×C4#93211yescharacteristic
325C3×C4C2×#21611yescharacteristic
326C3×C4#95011yescharacteristic
327C3×C4#95111yescharacteristic
328Dic3—41no—
329Dic3—41no—
330Dic3—41no—
331Dic3—122no—
332Dic3—122no—
333Dic3—242no—
334C32#164411yescharacteristic, complemented, Sylow
335C23—61no—
336Q8—41no—
337Q8—41no—
338Q8—61no—
339Q8—61no—
340Q8—362no—
341D4—32no—
342D4—61no—
343C2×C4—41no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—182no—
350C2×C4—722no—
351C2×C4D3×#6711yescharacteristic
352C2×C4D3×#6711yescharacteristic
353C2×C4#381111yescharacteristic
354C8—121no—
355C8—121no—
356C8—361no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—122no—
361C2×C3#503711yescharacteristic
362C2×C3#505011yescharacteristic
363C2×C3#515811yescharacteristic
364C2×C3#518611yescharacteristic
365C2×C3#520211yescharacteristic
366C2×C3#520211yescharacteristic
367D3—42no—
368C22—62no—
369C22—121no—
370C22—121no—
371C22#6398311yescharacteristic
372C4—21no—
373C4—81no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4—362no—
378C4#6393211yescharacteristic
379C4#6393311yescharacteristic
380C4D3×#21611yescharacteristic
381C4#6394311yescharacteristic
382C3—21no—
383C3#7106211yescharacteristic, complemented
384C3#7111311yescharacteristic, complemented
385C2—122no—
386C2#1059151911yescharacteristic
387C2#1059165311yescharacteristic
388C2#1059165811yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1644
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. #10592792C2semidirect
3. C3×#5090C2semidirect
5. #10589096C2semidirect
6. #10589083C2semidirect
34. #5090D3semidirect
117. C3×D12Q16semidirect
255. D12#216semidirect
334. C32#1644semidirect
383. C3#71062semidirect
384. C3#71113semidirect

Maximal subgroups

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

subgroupindex
#105939602
#105927922
C3×#50902
C3×#50872
#105890962
#105890832
#105890802
#711133
#710623