Finite Group Databasebeta

← #11779015

Subgroups — #11779015

Order 1152.

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

normal subgroups only

This group has 504 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, complemented
2C3×#5085C211yesmaximal, characteristic, complemented
3#10590364C211yesmaximal, characteristic, complemented
4#10590347C211yesmaximal, characteristic
5#10589412C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71923—31nomaximal
9#71264—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C3×#946—22no—
12C2×C3×#215—22no—
13C3×#897—21no—
14C3×#893—21no—
15C3×D3×C4⋊C4—21no—
16C3×#890—21no—
17C3×#874C2211yescharacteristic
18C3×C4×D12C2211yescharacteristic
19C3×C4×Dic6C2211yescharacteristic
20C2×#3715—21no—
21C2×#3713—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25#63562—21no—
26#63562—21no—
27C3×#554—31no—
28#5730D311yescharacteristic, complemented
29C3×#492—31no—
30#5085D311yescharacteristic, complemented
31#5075—31no—
32#5072—31no—
33#5027—31no—
34#5015—31no—
35#4546—31no—
36#4546—31no—
37#4546—61no—
38#4535—31no—
39D4⋊(C3⋊C8)—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8D412yes—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C2×C3×D12D411yescharacteristic
47C2×C3×C4×D3—21no—
48C2×C3×C4×D3—22no—
49C2×C3×Dic6D411yescharacteristic
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C32×C42C2311yescharacteristic
53C2×C3⋊(C3⋊C8)—21no—
54C2×C3⋊(C3⋊C8)—21no—
55C3×#215—44no—
56C3×(C2×D3)⋊C4—22no—
57C3×(C2×D3)⋊C4—21no—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3D411yescharacteristic
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—22no—
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C2×C3×C3⋊C8—21no—
65#3715—42no—
66#3713—42no—
67#1622—91noSylow
68C2×C3×SD16—62no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8C2×D311yescharacteristic
71C3×C4⋊Q8C2×D311yescharacteristic
72C3×#127—31no—
73C3×#125—61no—
74C3×C22⋊Q8—61no—
75C3×C22⋊Q8—62no—
76C3×C4×Q8—31no—
77C3×C4×D4—31no—
78C3×C4×D4—61no—
79C2×C3×C4⋊C4—61no—
80#946—22no—
81C2×#215—22no—
82C2×#215—61no—
83C2×#214—61no—
84C2×#213—61no—
85C2×#213—61no—
86#897—21no—
87#893—21no—
88D3×C4⋊C4—21no—
89#890—21no—
90#874C2×D311yescharacteristic
91C4×D12C2×D311yescharacteristic
92C4×Dic6C2×D311yescharacteristic
93C3×C4×C8—31no—
94#832—61no—
95#832—61no—
96#809—61no—
97#809—61no—
98C4⋊(C3⋊C8)—31no—
99C4⋊(C3⋊C8)—31no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—61no—
103C4⋊(C3⋊C8)—61no—
104C4×C3⋊C8C2×D311yescharacteristic
105C22×C3×D3—21no—
106C22×C3×D3—41no—
107C32×Q8—42no—
108C2×C32×C4—22no—
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C3×D12—21no—
116C3×D12—42no—
117C3×D12—41no—
118C3×D12—41no—
119C3×D12—41no—
120C3×D12—41no—
121C3×D12SD1612yescomplemented
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124C3×C4×D3—82no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C3⋊(C3⋊C8)—41no—
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—22no—
130#554—31no—
131#492—31no—
132#460—91no—
133#459—91no—
134#333—91no—
135Q8⋊C8—91no—
136D4⋊C8—91no—
137C2×C3×Q8—22no—
138C2×C3×Q8—31no—
139C2×C3×Q8#6712yes—
140C2×C3×Q8#6712yes—
141C2×C3×D4—31no—
142C2×C3×D4—61no—
143C2×C3×D4—61no—
144C2×C3×D4—61no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12#6711yescharacteristic
151C2×C4×D3—21no—
152C2×C4×D3—22no—
153C2×Dic6#6711yescharacteristic
154C3×SD16—124no—
155C2×C3×C8—61no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—62no—
166C3×C4⋊C4—61no—
167C3×C22⋊C4—62no—
168C3×C22⋊C4—61no—
169C3×C42—21no—
170C3×C42—61no—
171C3×C42C22×D311yescharacteristic
172C3×C42C22×D311yescharacteristic
173#215—44no—
174#215—122no—
175#214—122no—
176#213—122no—
177#213—122no—
178(C2×D3)⋊C4—22no—
179(C2×D3)⋊C4—21no—
180C4⋊Dic3—22no—
181C4⋊Dic3#6711yescharacteristic
182C4⋊Dic3#6711yescharacteristic
183Dic3⋊C4—22no—
184Dic3⋊C4—21no—
185C4×Dic3—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—61no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—121no—
192C2×C3⋊C8—121no—
193C22×C32#12011yescharacteristic
194C2×C3×D3—22no—
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C2×C3×D3—81no—
198C2×C3×D3—81no—
199C2×C3×D3—81no—
200C32×C4—21no—
201C32×C4—42no—
202C32×C4C2×SD1611yescharacteristic
203C32×C4#13511yescharacteristic
204C32×C4AGL(1,ℤ/8)11yescharacteristic
205C32×C4#13711yescharacteristic
206C3×Dic3—41no—
207C3×Dic3—41no—
208C3×Dic3—41no—
209C2×SD16—62no—
210C2×SD16—181no—
211C2×D8—181no—
212C4⋊Q8D3211yescharacteristic
213#127—31no—
214#125—61no—
215C22⋊Q8—61no—
216C22⋊Q8—62no—
217C4×Q8—31no—
218C4×D4—31no—
219C4×D4—61no—
220C2×C4⋊C4—61no—
221C4⋊C8—91no—
222C4⋊C8—91no—
223#102—181no—
224#102—181no—
225C4×C8—31no—
226C23×C3—61no—
227C23×C3—121no—
228C22×D3—21no—
229C22×D3—41no—
230C3×Q8—42no—
231C3×Q8—42no—
232C3×Q8—61no—
233C3×Q8—61no—
234C3×Q8—82no—
235C3×D4—32no—
236C3×D4—61no—
237C3×D4—122no—
238C3×D4—121no—
239C3×D4—121no—
240C3×D4—121no—
241C3×D4—121no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—42no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4—242no—
254C2×C3×C4D3×D411yescharacteristic
255C2×C3×C4D3×D411yescharacteristic
256C2×C3×C4C2×#6711yescharacteristic
257C2×C3×C4C2×#6711yescharacteristic
258C2×C3×C4C2×#6711yescharacteristic
259C2×C3×C4C2×#6711yescharacteristic
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263D12—21no—
264D12—42no—
265D12—41no—
266D12—41no—
267D12—41no—
268D12—41no—
269D12#21512yescomplemented
270C4×D3—41no—
271C4×D3—41no—
272C4×D3—82no—
273Dic6—21no—
274Dic6—21no—
275C3×C8—62no—
276C3⋊C8—22no—
277C3⋊C8—121no—
278C3⋊C8—121no—
279C3⋊C8—121no—
280C3⋊C8—121no—
281C3⋊C8—241no—
282C3⋊C8—241no—
283C2×C32#44811yescharacteristic
284C2×C32#44911yescharacteristic
285C2×C32#45311yescharacteristic
286C3×D3—42no—
287C3×D3—161no—
288C2×Q8—31no—
289C2×Q8#62512yes—
290C2×D4—31no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—61no—
294C22×C4—62no—
295C22×C4—61no—
296SD16—124no—
297SD16—362no—
298D8—362no—
299C2×C8—61no—
300C2×C8—181no—
301C2×C8—181no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—31no—
305C4⋊C4—31no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—61no—
309C22⋊C4—62no—
310C22⋊C4—61no—
311C42—61no—
312C42C2×D3211yescharacteristic
313C22×C3—21no—
314C22×C3—62no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—241no—
318C22×C3—241no—
319C22×C3—241no—
320C22×C3#93711yescharacteristic
321C22×C3#95311yescharacteristic
322C2×D3—22no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—82no—
338C3×C4—121no—
339C3×C4—121no—
340C3×C4—121no—
341C3×C4D3×SD1611yescharacteristic
342C3×C4#91411yescharacteristic
343C3×C4#91511yescharacteristic
344C3×C4#91611yescharacteristic
345C3×C4C2×#21511yescharacteristic
346C3×C4#94211yescharacteristic
347C3×C4#94911yescharacteristic
348C3×C4#95011yescharacteristic
349Dic3—41no—
350Dic3—41no—
351Dic3—41no—
352C32#162211yescharacteristic, complemented, Sylow
353C23—61no—
354C23—121no—
355Q8—42no—
356Q8—61no—
357Q8—61no—
358D4—32no—
359D4—61no—
360D4—122no—
361D4—121no—
362D4—121no—
363D4—121no—
364D4—121no—
365C2×C4—22no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—242no—
372C2×C4C2×#62511yescharacteristic
373C2×C4D3×#6711yescharacteristic
374C2×C4D3×#6711yescharacteristic
375C8—62no—
376C8—361no—
377C8—361no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—122no—
382C2×C3—481no—
383C2×C3#507711yescharacteristic
384C2×C3#517011yescharacteristic
385C2×C3#517211yescharacteristic
386C2×C3#522711yescharacteristic
387C2×C3#523811yescharacteristic
388C2×C3#524111yescharacteristic
389D3—42no—
390D3—161no—
391C22—62no—
392C22—121no—
393C22—121no—
394C22—241no—
395C22—241no—
396C22—241no—
397C22#6397611yescharacteristic
398C4—21no—
399C4—42no—
400C4—121no—
401C4—121no—
402C4—121no—
403C4D3×#21511yescharacteristic
404C4#6393911yescharacteristic
405C4#6394611yescharacteristic
406C4#6394711yescharacteristic
407C3—21no—
408C3#7126411yescharacteristic, complemented
409C3#7192311yescharacteristic, complemented
410C2—122no—
411C2—481no—
412C2#1059150911yescharacteristic
413C2#1059158311yescharacteristic
414C2#1059158611yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10590347C2
hypercenterC4⋊Q8D32
hypocenterC32#1622
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
1. C3×#5730C2semidirect
2. C3×#5085C2semidirect
3. #10590364C2semidirect
5. #10589412C2semidirect
6. #10589084C2semidirect
7. #10589081C2semidirect
28. #5730D3semidirect
30. #5085D3semidirect
121. C3×D12SD16semidirect
269. D12#215semidirect
352. C32#1622semidirect
408. C3#71264semidirect
409. C3#71923semidirect

Maximal subgroups

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

subgroupindex
C3×#57302
C3×#50852
#105903642
#105903472
#105894122
#105890842
#105890812
#719233
#712643