Finite Group Databasebeta

← #11780196

Subgroups — #11780196

Order 1152.

522 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 92 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#10593904C211yesmaximal, characteristic
2#10592789C211yesmaximal, characteristic, complemented
3C3×#5087C211yesmaximal, characteristic
4C3×#5078C211yesmaximal, characteristic, complemented
5#10589092C211yesmaximal, characteristic, complemented
6#10589085C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic
8#71376—31nomaximal
9#71114—31nomaximal
10C2×D3×Dic6—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C2×#3756—21no—
13C4×#634C2211yescharacteristic
14C3×C4×D12C2211yescharacteristic
15C3×#869C2211yescharacteristic
16#63919—21no—
17#63899—22no—
18#63898—21no—
19#63856—21no—
20#63662—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#809—21no—
23C3×#807—22no—
24C3×#807—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5727—31no—
28D4×Dic6—31no—
29C3×#500—31no—
30C3×#474—31no—
31#5090—31no—
32#5090—31no—
33#5090—61no—
34#5087D311yescharacteristic
35#5078D311yescharacteristic, complemented
36#4547—31no—
37D4⋊(C3⋊C8)—31no—
38#4486—31no—
39#4484—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8—21no—
43C2×#634D411yescharacteristic
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×Dic6—21no—
47C2×#627—22no—
48C2×D3×Dic3—22no—
49D3×Dic6—42no—
50D3×Dic6—42no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C42C2311yescharacteristic
54#3756—42no—
55C4⋊(C3⋊Dic3)D411yescharacteristic
56#3750—21no—
57C4×C3⋊Dic3—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3D412yes—
62C3×C4⋊Dic3D411yescharacteristic
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65#3724—22no—
66(C2×D3)⋊Dic3—22no—
67#1649—91noSylow
68C2×D3×Q8—61no—
69C22×Dic6—61no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4×D4—31no—
75#946—61no—
76C2×#216—61no—
77C2×#216—61no—
78C2×#216—62no—
79D4×Dic3—61no—
80#924—61no—
81#896—62no—
82#893—61no—
83#888—61no—
84#888—61no—
85#878—62no—
86C4×D12C2×D311yescharacteristic
87#869C2×D311yescharacteristic
88C4×Dic6—31no—
89C4×Dic6—31no—
90C4×Dic6—61no—
91C3×#107—61no—
92C3×#107—62no—
93C3×C4⋊C8—31no—
94C3×#102—61no—
95C3×C4×C8—31no—
96#835—61no—
97#835—61no—
98#835—62no—
99#809—21no—
100#807—22no—
101#807—21no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—61no—
105C4⋊(C3⋊C8)C2×D311yescharacteristic
106C4×C3⋊C8C2×D311yescharacteristic
107C22×C3×D3—21no—
108C32×Q8—41no—
109C32×Q8—41no—
110C2×C32×C4—41no—
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—21no—
116#634—21no—
117#634D812yes—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—21no—
120C3×D12—21no—
121C3×D12Q1612yescomplemented
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124C3×Dic6—41no—
125C3×Dic6—41no—
126#627—44no—
127D3×Dic3—44no—
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—41no—
131D4×Q8—91no—
132#500—31no—
133#474—31no—
134#462—91no—
135#332—91no—
136Q8⋊C8—91no—
137D4⋊C8—91no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—61no—
142C2×C3×D4—31no—
143C22×C3×C4—61no—
144C22×Dic3—62no—
145D3×Q8—122no—
146D3×Q8—122no—
147C2×D12#6711yescharacteristic
148C2×C4×D3—21no—
149C2×C4×D3—62no—
150C2×Dic6—21no—
151C2×Dic6—31no—
152C2×Dic6—31no—
153C2×Dic6—61no—
154C2×Dic6—62no—
155C2×Dic6—62no—
156C2×Dic6—122no—
157C2×Dic6—122no—
158C2×C3×C8—61no—
159C2×C3×C8—61no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—41no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—61no—
169C3×C22⋊C4—61no—
170C3×C42—21no—
171C3×C42C22×D311yescharacteristic
172C3×C42C22×D311yescharacteristic
173#217—62no—
174#216—122no—
175#216—122no—
176#216—124no—
177(C2×D3)⋊C4—21no—
178(C2×D3)⋊C4—62no—
179C4⋊Dic3—21no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—61no—
183C4⋊Dic3#6712yes—
184C4⋊Dic3#6711yescharacteristic
185Dic3⋊C4—62no—
186Dic3⋊C4—62no—
187Dic3⋊C4—61no—
188Dic3⋊C4—61no—
189Dic3⋊C4—121no—
190C4×Dic3—61no—
191C4×Dic3—61no—
192C4×Dic3—121no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—62no—
196C2×C3⋊C8—61no—
197C2×C3⋊C8—61no—
198C22×C32#12011yescharacteristic
199C2×C3×D3—22no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C32×C4—21no—
203C32×C4—81no—
204C32×C4C2×D811yescharacteristic
205C32×C4C2×Q1611yescharacteristic
206C32×C4AGL(1,ℤ/8)11yescharacteristic
207C32×C4#13711yescharacteristic
208C3⋊Dic3—22no—
209C3⋊Dic3—41no—
210C3×Dic3—42no—
211C3×Dic3—41no—
212C22×Q8—181no—
213C2×Q16—181no—
214C4⋊Q8—31no—
215C4⋊Q8—181no—
216C4⋊Q8D3211yescharacteristic
217C22⋊Q8—182no—
218C22⋊Q8—181no—
219C4×Q8—91no—
220C4×D4—31no—
221C4×D4—181no—
222#107—61no—
223#107—62no—
224C4⋊C8—31no—
225C4⋊C8—91no—
226#103—181no—
227#102—61no—
228C4×C8—31no—
229C23×C3—61no—
230C22×D3—21no—
231C3×Q8—42no—
232C3×Q8—42no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—121no—
238C3×Q8—121no—
239C3×D4—32no—
240C3×D4—61no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—61no—
248C2×C3×C4—62no—
249C2×C3×C4—121no—
250C2×C3×C4—121no—
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4D3×D411yescharacteristic
254C2×C3×C4D3×D411yescharacteristic
255C2×C3×C4C2×#6711yescharacteristic
256C2×C3×C4C2×#6711yescharacteristic
257C2×Dic3—21no—
258C2×Dic3—22no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—121no—
264C2×Dic3—121no—
265C2×Dic3—124no—
266D12—21no—
267D12#21612yescomplemented
268C4×D3—41no—
269C4×D3—41no—
270C4×D3—124no—
271Dic6—32no—
272Dic6—32no—
273Dic6—41no—
274Dic6—41no—
275Dic6—62no—
276Dic6—61no—
277Dic6—61no—
278Dic6—121no—
279Dic6—124no—
280Dic6—124no—
281C3×C8—62no—
282C3×C8—121no—
283C3⋊C8—22no—
284C3⋊C8—41no—
285C3⋊C8—122no—
286C3⋊C8—121no—
287C3⋊C8—121no—
288C2×C32#44711yescharacteristic
289C2×C32#45111yescharacteristic
290C2×C32#45611yescharacteristic
291C3×D3—42no—
292C2×Q8—21no—
293C2×Q8—61no—
294C2×Q8—91no—
295C2×Q8—182no—
296C2×Q8—362no—
297C2×Q8—362no—
298C2×D4—31no—
299C22×C4—61no—
300C22×C4—182no—
301Q16—362no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—181no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—31no—
308C4⋊C4—32no—
309C4⋊C4—61no—
310C4⋊C4—91no—
311C4⋊C4—182no—
312C4⋊C4—181no—
313C22⋊C4—61no—
314C22⋊C4—182no—
315C42—181no—
316C42C2×D3211yescharacteristic
317C22×C3—21no—
318C22×C3—62no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3#93711yescharacteristic
322C22×C3#93711yescharacteristic
323C2×D3—22no—
324C2×D3—41no—
325C2×D3—41no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—41no—
333C3×C4—82no—
334C3×C4—81no—
335C3×C4—81no—
336C3×C4—121no—
337C3×C4—122no—
338C3×C4D3×D811yescharacteristic
339C3×C4D3×D811yescharacteristic
340C3×C4#91111yescharacteristic
341C3×C4#91111yescharacteristic
342C3×C4D3×Q1611yescharacteristic
343C3×C4#91811yescharacteristic
344C3×C4C2×#21611yescharacteristic
345C3×C4#95111yescharacteristic
346Dic3—42no—
347Dic3—41no—
348Dic3—62no—
349Dic3—62no—
350Dic3—122no—
351Dic3—121no—
352Dic3—121no—
353Dic3—241no—
354C32#164911yescharacteristic, complemented, Sylow
355C23—61no—
356Q8—41no—
357Q8—41no—
358Q8—92no—
359Q8—121no—
360Q8—121no—
361Q8—181no—
362Q8—364no—
363D4—32no—
364D4—61no—
365C2×C4—41no—
366C2×C4—61no—
367C2×C4—62no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—364no—
373C2×C4D3×#6711yescharacteristic
374C2×C4D3×#6711yescharacteristic
375C2×C4#381111yescharacteristic
376C8—62no—
377C8—121no—
378C8—361no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—122no—
383C2×C3#505011yescharacteristic
384C2×C3#509511yescharacteristic
385C2×C3#515611yescharacteristic
386C2×C3#515611yescharacteristic
387C2×C3#518611yescharacteristic
388C2×C3#520511yescharacteristic
389D3—42no—
390C22—62no—
391C22—121no—
392C22—121no—
393C22#6398311yescharacteristic
394C4—21no—
395C4—81no—
396C4—121no—
397C4—122no—
398C4—182no—
399C4—361no—
400C4#6392611yescharacteristic
401C4#6392711yescharacteristic
402C4D3×#21611yescharacteristic
403C4#6394311yescharacteristic
404C3—21no—
405C3#7111411yescharacteristic, complemented
406C3#7137611yescharacteristic, complemented
407C2—122no—
408C2#1059151611yescharacteristic
409C2#1059165311yescharacteristic
410C2#1059166111yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1649
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. #10592789C2semidirect
4. C3×#5078C2semidirect
5. #10589092C2semidirect
6. #10589085C2semidirect
35. #5078D3semidirect
121. C3×D12Q16semidirect
267. D12#216semidirect
354. C32#1649semidirect
405. C3#71114semidirect
406. C3#71376semidirect

Maximal subgroups

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

subgroupindex
#105939042
#105927892
C3×#50872
C3×#50782
#105890922
#105890852
#105890812
#713763
#711143