Finite Group Databasebeta

← #11780289

Subgroups — #11780289

Order 1152.

517 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 94 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
1Dic6×D12C211yesmaximal, characteristic
2#10592779C211yesmaximal, characteristic, complemented
3C3×#5094C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic
5#10589094C211yesmaximal, characteristic, complemented
6#10589083C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic
8#71127—31nomaximal
9#71050—31nomaximal
10C2×D3×Dic6—21no—
11C32×C4⋊Q8C2211yescharacteristic
12#64078C2211yescharacteristic
13C2×C3×#215—21no—
14C3×C4×D12C2211yescharacteristic
15C3×C4×Dic6C2211yescharacteristic
16#63918—21no—
17#63893—22no—
18Dic3×D12—21no—
19#63857—21no—
20#63635—22no—
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#835—21no—
23C3×#810—21no—
24C3×#808—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4⋊(C3⋊C8)C2211yescharacteristic
27Q8×D12—31no—
28D4×Dic6—31no—
29C3×#475—31no—
30C3×#461—31no—
31#5094D311yescharacteristic, complemented
32#5083D311yescharacteristic
33#5080—31no—
34#5080—31no—
35#5080—61no—
36#4535—31no—
37D4⋊(C3⋊C8)—31no—
38#4486—31no—
39#4484—31no—
40#4483—31no—
41#4482—31no—
42C2×C32×Q8—21no—
43C2×#634—21no—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×Dic6D411yescharacteristic
47C2×#627—22no—
48C2×D3×Dic3—22no—
49D3×Dic6—42no—
50D3×Dic6—42no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C42C2311yescharacteristic
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)D412yes—
56C2×C3⋊(C3⋊C8)—21no—
57C3×#215—42no—
58C3×(C2×D3)⋊C4—21no—
59C3×C4⋊Dic3D411yescharacteristic
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—21no—
62C3×C4×Dic3—21no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65Dic3⋊Dic3—22no—
66(C2×D3)⋊Dic3—22no—
67#1651—91noSylow
68C2×D3×Q8—61no—
69C22×Dic6—61no—
70C2×C3×SD16—61no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4×Q8—31no—
75C3×C4×D4—31no—
76C2×#215—21no—
77D4×Dic3—61no—
78#924—61no—
79#897—62no—
80#897—61no—
81#888—61no—
82#878—62no—
83C4×D12—61no—
84C4×D12C2×D311yescharacteristic
85#869—31no—
86#869—31no—
87#869—61no—
88#869—61no—
89C4×Dic6C2×D311yescharacteristic
90C3×#106—61no—
91C3×C4⋊C8—31no—
92C3×C4⋊C8—31no—
93C3×#103—61no—
94C3×#103—61no—
95#835—21no—
96#810—21no—
97#808—21no—
98#808—62no—
99#808—62no—
100#808—64no—
101C4⋊(C3⋊C8)C2×D311yescharacteristic
102C4⋊(C3⋊C8)C2×D311yescharacteristic
103C4×C3⋊C8—31no—
104C4×C3⋊C8—31no—
105C4×C3⋊C8—61no—
106C22×C3×D3—21no—
107C32×Q8—41no—
108C32×Q8—41no—
109C2×C32×C4—41no—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C3⋊Dic3—22no—
114#634—41no—
115#634—41no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C3×D12—21no—
120C3×D12SD1612yes—
121C3×C4×D3—41no—
122C3×C4×D3—41no—
123C3×Dic6—21no—
124C3×Dic6SD1612yescomplemented
125#627—44no—
126D3×Dic3—44no—
127C3⋊(C3⋊C8)—22no—
128C3×C3⋊C8—41no—
129C3×C3⋊C8—41no—
130D4×Q8—91no—
131#498—91no—
132#475—31no—
133#461—31no—
134#332—91no—
135Q8⋊C8—91no—
136D4⋊C8—91no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—31no—
140C2×C3×Q8—41no—
141C2×C3×D4—31no—
142C22×C3×C4—61no—
143C22×Dic3—62no—
144D3×Q8—122no—
145D3×Q8—122no—
146C2×D12#6711yescharacteristic
147C2×C4×D3—21no—
148C2×C4×D3—62no—
149C2×Dic6—62no—
150C2×Dic6—62no—
151C2×Dic6—61no—
152C2×Dic6—61no—
153C2×Dic6—121no—
154C2×Dic6—122no—
155C2×Dic6—122no—
156C2×Dic6#6711yescharacteristic
157C3×SD16—122no—
158C2×C3×C8—61no—
159C2×C3×C8—61no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—61no—
169C3×C22⋊C4—61no—
170C3×C42—21no—
171C3×C42—61no—
172C3×C42C22×D311yescharacteristic
173C3×C42C22×D311yescharacteristic
174#217—62no—
175#215—42no—
176(C2×D3)⋊C4—21no—
177(C2×D3)⋊C4—62no—
178C4⋊Dic3—32no—
179C4⋊Dic3—32no—
180C4⋊Dic3—62no—
181C4⋊Dic3—62no—
182C4⋊Dic3—61no—
183C4⋊Dic3—61no—
184C4⋊Dic3—121no—
185C4⋊Dic3#6711yescharacteristic
186C4⋊Dic3#6711yescharacteristic
187Dic3⋊C4—21no—
188Dic3⋊C4—62no—
189C4×Dic3—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—62no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—61no—
195C22×C32#12011yescharacteristic
196C2×C3×D3—22no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C32×C4—21no—
200C32×C4—81no—
201C32×C4C2×SD1611yescharacteristic
202C32×C4C2×SD1611yescharacteristic
203C32×C4AGL(1,ℤ/8)11yescharacteristic
204C32×C4#13711yescharacteristic
205C3⋊Dic3—42no—
206C3×Dic3—22no—
207C3×Dic3—41no—
208C3×Dic3—41no—
209C22×Q8—181no—
210C2×SD16—61no—
211C4⋊Q8—91no—
212C4⋊Q8—181no—
213C4⋊Q8D3211yescharacteristic
214C22⋊Q8—182no—
215C22⋊Q8—181no—
216C4×Q8—31no—
217C4×D4—31no—
218C4×D4—181no—
219#106—61no—
220#106—182no—
221C4⋊C8—31no—
222C4⋊C8—31no—
223#103—61no—
224#103—61no—
225C4×C8—91no—
226C23×C3—61no—
227C22×D3—21no—
228C3×Q8—32no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—61no—
234C3×Q8—81no—
235C3×Q8—81no—
236C3×D4—32no—
237C3×D4—61no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—121no—
248C2×C3×C4—121no—
249C2×C3×C4D3×D411yescharacteristic
250C2×C3×C4D3×D411yescharacteristic
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4C2×#6711yescharacteristic
254C2×C3×C4C2×#6711yescharacteristic
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—122no—
261C2×Dic3—124no—
262D12—21no—
263D12#21412yes—
264C4×D3—41no—
265C4×D3—41no—
266C4×D3—124no—
267Dic6—21no—
268Dic6—124no—
269Dic6—124no—
270Dic6—121no—
271Dic6—121no—
272Dic6—121no—
273Dic6—121no—
274Dic6—241no—
275Dic6—241no—
276Dic6#21512yescomplemented
277C3×C8—121no—
278C3×C8—121no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C3⋊C8—64no—
282C3⋊C8—62no—
283C3⋊C8—62no—
284C2×C32#44811yescharacteristic
285C2×C32#45011yescharacteristic
286C2×C32#45611yescharacteristic
287C3×D3—42no—
288C2×Q8—21no—
289C2×Q8—31no—
290C2×Q8—182no—
291C2×Q8—181no—
292C2×Q8—362no—
293C2×Q8—362no—
294C2×D4—31no—
295C22×C4—61no—
296C22×C4—182no—
297SD16—122no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—181no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—31no—
304C4⋊C4—31no—
305C4⋊C4—61no—
306C4⋊C4—92no—
307C4⋊C4—182no—
308C4⋊C4—181no—
309C22⋊C4—61no—
310C22⋊C4—182no—
311C42—61no—
312C42C2×D3211yescharacteristic
313C22×C3—21no—
314C22×C3—62no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3#93711yescharacteristic
318C22×C3#93711yescharacteristic
319C2×D3—22no—
320C2×D3—41no—
321C2×D3—41no—
322C3×C4—22no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—62no—
330C3×C4—82no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4—121no—
334C3×C4—121no—
335C3×C4D3×SD1611yescharacteristic
336C3×C4D3×SD1611yescharacteristic
337C3×C4#91411yescharacteristic
338C3×C4#91511yescharacteristic
339C3×C4C2×#21411yescharacteristic
340C3×C4C2×#21511yescharacteristic
341C3×C4#94911yescharacteristic
342C3×C4#95111yescharacteristic
343Dic3—22no—
344Dic3—41no—
345Dic3—41no—
346Dic3—122no—
347Dic3—122no—
348Dic3—242no—
349C32#165111yescharacteristic, complemented, Sylow
350C23—61no—
351Q8—32no—
352Q8—41no—
353Q8—41no—
354Q8—61no—
355Q8—364no—
356Q8—361no—
357Q8—361no—
358D4—32no—
359D4—61no—
360C2×C4—41no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—182no—
367C2×C4—364no—
368C2×C4D3×#6711yescharacteristic
369C2×C4D3×#6711yescharacteristic
370C2×C4#381111yescharacteristic
371C8—121no—
372C8—121no—
373C8—182no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—122no—
378C2×C3#504011yescharacteristic
379C2×C3#504611yescharacteristic
380C2×C3#516911yescharacteristic
381C2×C3#517011yescharacteristic
382C2×C3#520511yescharacteristic
383C2×C3#520511yescharacteristic
384D3—42no—
385C22—62no—
386C22—121no—
387C22—121no—
388C22#6398311yescharacteristic
389C4—21no—
390C4—62no—
391C4—81no—
392C4—121no—
393C4—121no—
394C4—362no—
395C4D3×#21411yescharacteristic
396C4#6392911yescharacteristic
397C4D3×#21511yescharacteristic
398C4#6393911yescharacteristic
399C3—21no—
400C3#7105011yescharacteristic, complemented
401C3#7112711yescharacteristic, complemented
402C2—122no—
403C2#1059151711yescharacteristic
404C2#1059165111yescharacteristic
405C2#1059166011yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1651
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. #10592779C2semidirect
3. C3×#5094C2semidirect
5. #10589094C2semidirect
6. #10589083C2semidirect
31. #5094D3semidirect
124. C3×Dic6SD16semidirect
276. Dic6#215semidirect
349. C32#1651semidirect
400. C3#71050semidirect
401. C3#71127semidirect

Maximal subgroups

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

subgroupindex
Dic6×D122
#105927792
C3×#50942
C3×#50832
#105890942
#105890832
#105890802
#711273
#710503