Finite Group Databasebeta

← #11779914

Subgroups — #11779914

Order 1152.

465 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 96 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×#5730C211yesmaximal, characteristic
2#10592778C211yesmaximal, characteristic, complemented
3#10590365C211yesmaximal, characteristic
4#10590301C211yesmaximal, characteristic, complemented
5C3×#4546C211yesmaximal, characteristic, complemented
6#10589083C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic
8#71925—31nomaximal
9#68494—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×#946—22no—
13C3×#897—21no—
14C3×#893—21no—
15C3×D3×C4⋊C4—21no—
16C3×#890—21no—
17C3×C4×D12C2211yescharacteristic
18C3×C4×Dic6C2211yescharacteristic
19C2×#3716—21no—
20#63662—22no—
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24#63573—21no—
25#63567—21no—
26#63566—21no—
27C3×#554—31no—
28#5730D311yescharacteristic
29#5079—31no—
30#5079—31no—
31#5079—61no—
32#5020—31no—
33#5019—31no—
34#4851—31no—
35#4841—31no—
36C3×#333—31no—
37#4546D311yescharacteristic, complemented
38#4535—31no—
39D4⋊(C3⋊C8)—31no—
40#4484—31no—
41#4483—31no—
42C2×C32×Q8D412yes—
43C2×#634—21no—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3—22no—
47C2×C3×Dic6D411yescharacteristic
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C42C2311yescharacteristic
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)—21no—
53C2×C3⋊(C3⋊C8)—21no—
54C3×(C2×D3)⋊C4—22no—
55C3×(C2×D3)⋊C4—21no—
56C3×C4⋊Dic3—22no—
57C3×C4⋊Dic3D411yescharacteristic
58C3×C4⋊Dic3D411yescharacteristic
59C3×Dic3⋊C4—22no—
60C3×Dic3⋊C4—21no—
61C3×C4×Dic3—21no—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8—21no—
64#3716—42no—
65#1643—91noSylow
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8C2×D311yescharacteristic
68C3×C4⋊Q8C2×D311yescharacteristic
69C3×#125—61no—
70C3×C22⋊Q8—61no—
71C3×C22⋊Q8—62no—
72C3×C4×Q8—31no—
73C3×C4×D4—31no—
74C3×C4×D4—61no—
75C2×C3×C4⋊C4—61no—
76#946—22no—
77C2×#214—61no—
78C2×#204—61no—
79#897—21no—
80#893—21no—
81D3×C4⋊C4—21no—
82#890—21no—
83C4×D12C2×D311yescharacteristic
84#869—31no—
85#869—31no—
86#869—61no—
87C4×Dic6C2×D311yescharacteristic
88C3×C4⋊C8—31no—
89C3×C4⋊C8—31no—
90#835—61no—
91#835—64no—
92#835—62no—
93#835—62no—
94#817—61no—
95#816—61no—
96#816—61no—
97#810—61no—
98#808—61no—
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—42no—
106C2×C32×C4—22no—
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C3⋊Dic3—41no—
111#634—41no—
112#634—41no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C3×D12—21no—
117C3×D12SD1612yes—
118C3×C4×D3—41no—
119C3×C4×D3—41no—
120C3×C4×D3—82no—
121C3×Dic6—21no—
122C3×Dic6—21no—
123C3⋊(C3⋊C8)—41no—
124C3×C3⋊C8—41no—
125C3×C3⋊C8—41no—
126#554—31no—
127#487—91no—
128#475—91no—
129#461—91no—
130#333—31no—
131Q8⋊C8—91no—
132D4⋊C8—91no—
133C2×C3×Q8—22no—
134C2×C3×Q8—31no—
135C2×C3×Q8D1212yes—
136C2×C3×Q8#6712yes—
137C2×C3×D4—31no—
138C22×C3×C4—62no—
139C22×C3×C4—61no—
140C2×D12#6711yescharacteristic
141C2×C4×D3—21no—
142C2×C4×D3—22no—
143C2×Dic6—61no—
144C2×Dic6—61no—
145C2×Dic6—121no—
146C2×Dic6#6711yescharacteristic
147C2×C3×C8—61no—
148C2×C3×C8—61no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—62no—
158C3×C4⋊C4—62no—
159C3×C4⋊C4—61no—
160C3×C22⋊C4—62no—
161C3×C22⋊C4—61no—
162C3×C42—21no—
163C3×C42—61no—
164C3×C42C22×D311yescharacteristic
165C3×C42C22×D311yescharacteristic
166#214—122no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—21no—
169C4⋊Dic3—22no—
170C4⋊Dic3—62no—
171C4⋊Dic3—61no—
172C4⋊Dic3—61no—
173C4⋊Dic3—61no—
174C4⋊Dic3—61no—
175C4⋊Dic3—121no—
176C4⋊Dic3#6711yescharacteristic
177C4⋊Dic3#6711yescharacteristic
178Dic3⋊C4—22no—
179Dic3⋊C4—21no—
180C4×Dic3—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—62no—
184C2×C3⋊C8—61no—
185C2×C3⋊C8—61no—
186#204—122no—
187C22×C32#12011yescharacteristic
188C2×C3×D3—22no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C32×C4—21no—
192C32×C4—42no—
193C32×C4C2×SD1611yescharacteristic
194C32×C4#13511yescharacteristic
195C32×C4#13711yescharacteristic
196C32×C4#13711yescharacteristic
197C3⋊Dic3—81no—
198C3×Dic3—41no—
199C3×Dic3—41no—
200C3×Dic3—41no—
201C2×SD16—181no—
202C4⋊Q8—91no—
203C4⋊Q8D3211yescharacteristic
204#125—61no—
205C22⋊Q8—61no—
206C22⋊Q8—62no—
207C4×Q8—31no—
208C4×D4—31no—
209C4×D4—61no—
210C2×C4⋊C4—61no—
211#106—181no—
212C4⋊C8—31no—
213C4⋊C8—31no—
214#103—181no—
215#103—181no—
216#103—182no—
217C4×C8—91no—
218C23×C3—61no—
219C22×D3—21no—
220C3×Q8—42no—
221C3×Q8—42no—
222C3×Q8—44no—
223C3×Q8—61no—
224C3×Q8—61no—
225C3×D4—32no—
226C3×D4—61no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—42no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—121no—
237C2×C3×C4—121no—
238C2×C3×C4—242no—
239C2×C3×C4C2×D1211yescharacteristic
240C2×C3×C4D3×D411yescharacteristic
241C2×C3×C4D3×D411yescharacteristic
242C2×C3×C4C2×#6711yescharacteristic
243C2×C3×C4C2×#6711yescharacteristic
244C2×C3×C4C2×#6711yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—122no—
249C2×Dic3—121no—
250C2×Dic3—121no—
251D12—21no—
252D12#21512yes—
253C4×D3—41no—
254C4×D3—41no—
255C4×D3—82no—
256Dic6—21no—
257Dic6—21no—
258Dic6—121no—
259Dic6—121no—
260Dic6—121no—
261Dic6—121no—
262Dic6—241no—
263Dic6—241no—
264C3×C8—121no—
265C3×C8—121no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C3⋊C8—122no—
269C3⋊C8—121no—
270C3⋊C8—121no—
271C2×C32#44811yescharacteristic
272C2×C32#45211yescharacteristic
273C2×C32#45511yescharacteristic
274C3×D3—42no—
275C2×Q8—31no—
276C2×Q8—181no—
277C2×Q8#62612yes—
278C2×D4—31no—
279C22×C4—62no—
280C22×C4—61no—
281SD16—362no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—181no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4—181no—
293C4⋊C4—181no—
294C22⋊C4—62no—
295C22⋊C4—61no—
296C42—61no—
297C42C2×D3211yescharacteristic
298C22×C3—21no—
299C22×C3—62no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3#88211yescharacteristic
303C22×C3#95311yescharacteristic
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—42no—
315C3×C4—42no—
316C3×C4—82no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—121no—
320C3×C4D3×SD1611yescharacteristic
321C3×C4#91611yescharacteristic
322C3×C4#91811yescharacteristic
323C3×C4#91811yescharacteristic
324C3×C4C2×#21511yescharacteristic
325C3×C4#94211yescharacteristic
326C3×C4#95011yescharacteristic
327C3×C4#95111yescharacteristic
328Dic3—41no—
329Dic3—41no—
330Dic3—41no—
331Dic3—242no—
332Dic3—241no—
333Dic3—241no—
334C32#164311yescharacteristic, complemented, Sylow
335C23—61no—
336Q8—42no—
337Q8—61no—
338Q8—61no—
339Q8—361no—
340Q8—361no—
341D4—32no—
342D4—61no—
343C2×C4—22no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—242no—
350C2×C4—361no—
351C2×C4C2×#62611yescharacteristic
352C2×C4D3×#6711yescharacteristic
353C2×C4D3×#6711yescharacteristic
354C8—121no—
355C8—121no—
356C8—361no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—122no—
361C2×C3#474611yescharacteristic
362C2×C3#480611yescharacteristic
363C2×C3#480811yescharacteristic
364C2×C3#522711yescharacteristic
365C2×C3#523911yescharacteristic
366C2×C3#524211yescharacteristic
367D3—42no—
368C22—62no—
369C22—121no—
370C22—121no—
371C22#6397711yescharacteristic
372C4—21no—
373C4—42no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4—721no—
378C4D3×#21511yescharacteristic
379C4#6394311yescharacteristic
380C4#6394711yescharacteristic
381C4#6394811yescharacteristic
382C3—21no—
383C3#6849411yescharacteristic, complemented
384C3#7192511yescharacteristic, complemented
385C2—122no—
386C2#1059151311yescharacteristic
387C2#1059158411yescharacteristic
388C2#1059159911yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1643
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. #10592778C2semidirect
4. #10590301C2semidirect
5. C3×#4546C2semidirect
6. #10589083C2semidirect
37. #4546D3semidirect
334. C32#1643semidirect
383. C3#68494semidirect
384. C3#71925semidirect

Maximal subgroups

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

subgroupindex
C3×#57302
#105927782
#105903652
#105903012
C3×#45462
#105890832
#105890802
#719253
#684943