Finite Group Databasebeta

← #11779864

Subgroups — #11779864

Order 1152.

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

normal subgroups only

This group has 560 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#10593900C211yesmaximal, characteristic
2#10592782C211yesmaximal, characteristic, complemented
3C3×#5094C211yesmaximal, characteristic, complemented
4C3×#5088C211yesmaximal, characteristic, complemented
5#10589090C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic, complemented
8#71199—31nomaximal
9#71124—31nomaximal
10C2×#3796—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C2×#3755—21no—
13C4×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×#874C2211yescharacteristic
15C3×C4×Dic6C2211yescharacteristic
16#63916—21no—
17#63911—21no—
18Dic3×D12—21no—
19#63886—22no—
20#63662—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#810—21no—
23C3×#809—22no—
24C3×#808—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5718—31no—
28#5577—31no—
29C3×#486—31no—
30C3×#475—31no—
31#5094D311yescharacteristic, complemented
32#5088D311yescharacteristic, complemented
33#5083—31no—
34#5083—31no—
35#5083—61no—
36#4544—31no—
37#4535—31no—
38#4485—31no—
39#4483—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12D412yes—
48C2×C3×Dic6D411yescharacteristic
49C2×#626—22no—
50C2×#626—22no—
51C2×D3×Dic3—22no—
52#3796—42no—
53#3796—42no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C42C2311yescharacteristic
57#3755—42no—
58#3752—21no—
59C4⋊(C3⋊Dic3)D411yescharacteristic
60C2×C3⋊(C3⋊C8)—21no—
61C3×C4⋊Dic3D411yescharacteristic
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66(C2×D3)⋊Dic3—22no—
67#1642—91noSylow
68C2×#239—61no—
69C2×#237—61no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8C2×D311yescharacteristic
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×#127—31no—
74C3×C4×Q8—31no—
75C2×#215—61no—
76C2×#215—61no—
77C2×#215—62no—
78#940—61no—
79#939—62no—
80#938—61no—
81D4×Dic3—61no—
82#895—61no—
83#895—62no—
84#874—61no—
85#874C2×D311yescharacteristic
86C4×D12—31no—
87C4×D12—31no—
88C4×D12—61no—
89C4×D12—61no—
90C4×Dic6C2×D311yescharacteristic
91C3×#106—61no—
92C3×C4⋊C8—31no—
93C3×#103—61no—
94C3×#102—62no—
95C3×C4×C8—31no—
96#835—61no—
97#835—61no—
98#835—62no—
99#810—21no—
100#809—22no—
101#808—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—
108C22×C3×D3—22no—
109C32×Q8—41no—
110C32×Q8—41no—
111C2×C32×C4—41no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C3⋊Dic3—21no—
116C4⋊(C3⋊D3)—21no—
117C4⋊(C3⋊D3)—21no—
118C4×C3⋊D3—41no—
119C4×C3⋊D3—41no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—21no—
122C3×D12—42no—
123C3×D12—41no—
124C3×D12—41no—
125C3×D12—41no—
126C3×D12—41no—
127C3×Dic6—21no—
128C3×Dic6SD1612yescomplemented
129#626—44no—
130#626—82no—
131D3×Dic3—44no—
132C3⋊(C3⋊C8)—41no—
133C3×C3⋊C8—41no—
134C3×C3⋊C8—41no—
135#550—91no—
136#486—31no—
137#475—31no—
138#461—91no—
139#331—91no—
140Q8⋊C8—91no—
141D4⋊C8—91no—
142C2×C3×Q8—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—31no—
146C2×C3×D4—32no—
147C2×C3×D4—61no—
148C2×C3×D4—61no—
149C2×#67—62no—
150C2×#67—62no—
151C22×Dic3—62no—
152#239—122no—
153#239—122no—
154#237—122no—
155#237—122no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12—31no—
159C2×D12—31no—
160C2×D12—61no—
161C2×D12—62no—
162C2×D12—62no—
163C2×D12#6712yes—
164C2×C4×D3—62no—
165C2×C4×D3—61no—
166C2×C4×D3—61no—
167C2×C4×D3—121no—
168C2×Dic6#6711yescharacteristic
169C2×C3×C8—61no—
170C2×C3×C8—61no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—41no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—61no—
179C3×C42—21no—
180C3×C42—61no—
181C3×C42C22×D311yescharacteristic
182C3×C42C22×D311yescharacteristic
183#217—62no—
184#215—124no—
185#215—122no—
186#215—122no—
187(C2×D3)⋊C4—62no—
188(C2×D3)⋊C4—61no—
189(C2×D3)⋊C4—61no—
190(C2×D3)⋊C4—121no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—61no—
194C4⋊Dic3#6711yescharacteristic
195Dic3⋊C4—21no—
196C4×Dic3—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—62no—
200C2×C3⋊C8—61no—
201C2×C3⋊C8—61no—
202C22×C32#12011yescharacteristic
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3×D3—42no—
207C2×C3×D3—42no—
208C2×C3×D3—42no—
209C32×C4—21no—
210C32×C4—81no—
211C32×C4C2×SD1611yescharacteristic
212C32×C4#13511yescharacteristic
213C32×C4AGL(1,ℤ/8)11yescharacteristic
214C32×C4AGL(1,ℤ/8)11yescharacteristic
215C3⋊Dic3—41no—
216C3×Dic3—22no—
217C3×Dic3—41no—
218C2×#41—181no—
219C2×SD16—181no—
220C4⋊Q8D3211yescharacteristic
221#127—31no—
222#127—181no—
223#121—181no—
224#121—182no—
225C4×Q8—31no—
226C4×D4—91no—
227C4×D4—181no—
228#106—61no—
229C4⋊C8—31no—
230C4⋊C8—91no—
231#103—61no—
232#103—181no—
233#102—62no—
234C4×C8—31no—
235C23×C3—62no—
236C22×D3—22no—
237C22×D3—61no—
238C22×D3—61no—
239C22×D3—121no—
240C3×Q8—32no—
241C3×Q8—42no—
242C3×Q8—42no—
243C3×Q8—41no—
244C3×Q8—41no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×Q8—61no—
248C3×D4—122no—
249C3×D4—121no—
250C3×D4—121no—
251C3×D4—121no—
252C3×D4—121no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4D3×D411yescharacteristic
262C2×C3×C4D3×D411yescharacteristic
263C2×C3×C4D3×D411yescharacteristic
264C2×C3×C4D3×D411yescharacteristic
265C2×C3×C4C2×#6711yescharacteristic
266C2×C3×C4C2×#6711yescharacteristic
267#67—124no—
268#67—242no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—121no—
274C2×Dic3—124no—
275D12—42no—
276D12—41no—
277D12—41no—
278D12—41no—
279D12—41no—
280D12—62no—
281D12—61no—
282D12—61no—
283D12—61no—
284D12—61no—
285D12—121no—
286D12—124no—
287D12—242no—
288C4×D3—124no—
289C4×D3—121no—
290C4×D3—121no—
291C4×D3—121no—
292C4×D3—121no—
293C4×D3—241no—
294C4×D3—241no—
295Dic6—21no—
296Dic6#21512yescomplemented
297C3×C8—121no—
298C3×C8—121no—
299C3⋊C8—41no—
300C3⋊C8—41no—
301C3⋊C8—122no—
302C3⋊C8—121no—
303C3⋊C8—121no—
304C2×C32#44911yescharacteristic
305C2×C32#45011yescharacteristic
306C2×C32#45511yescharacteristic
307C3⋊D3—81no—
308C3×D3—82no—
309#41—362no—
310#41—362no—
311C2×Q8—21no—
312C2×Q8—31no—
313C2×D4—32no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—91no—
317C2×D4—182no—
318C2×D4—182no—
319C22×C4—182no—
320C22×C4—181no—
321SD16—362no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8—181no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—31no—
328C4⋊C4—61no—
329C4⋊C4—91no—
330C22⋊C4—182no—
331C22⋊C4—181no—
332C42—61no—
333C42C2×D3211yescharacteristic
334C22×C3—21no—
335C22×C3—122no—
336C22×C3—122no—
337C22×C3—122no—
338C22×C3#93711yescharacteristic
339C22×C3#93711yescharacteristic
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—42no—
343C2×D3—122no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—241no—
351C2×D3—241no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—62no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4—81no—
363C3×C4—121no—
364C3×C4#91111yescharacteristic
365C3×C4#91211yescharacteristic
366C3×C4D3×SD1611yescharacteristic
367C3×C4#91411yescharacteristic
368C3×C4#91411yescharacteristic
369C3×C4#91611yescharacteristic
370C3×C4C2×#21511yescharacteristic
371C3×C4#94911yescharacteristic
372Dic3—22no—
373Dic3—41no—
374Dic3—121no—
375Dic3—121no—
376Dic3—241no—
377C32#164211yescharacteristic, complemented, Sylow
378C23—62no—
379C23—181no—
380Q8—32no—
381Q8—41no—
382Q8—41no—
383Q8—61no—
384D4—122no—
385D4—121no—
386D4—121no—
387D4—121no—
388D4—121no—
389D4—181no—
390D4—181no—
391D4—364no—
392D4—722no—
393C2×C4—41no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—181no—
397C2×C4—364no—
398C2×C4—361no—
399C2×C4—361no—
400C2×C4D3×#6711yescharacteristic
401C2×C4D3×#6711yescharacteristic
402C2×C4#381111yescharacteristic
403C8—121no—
404C8—121no—
405C8—361no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—242no—
410C2×C3#504611yescharacteristic
411C2×C3#506111yescharacteristic
412C2×C3#515811yescharacteristic
413C2×C3#516911yescharacteristic
414C2×C3#517211yescharacteristic
415C2×C3#520211yescharacteristic
416D3—82no—
417D3—242no—
418D3—241no—
419D3—241no—
420C22—122no—
421C22—122no—
422C22—122no—
423C22—361no—
424C22—361no—
425C22—361no—
426C22#6398311yescharacteristic
427C4—21no—
428C4—62no—
429C4—81no—
430C4—121no—
431C4—361no—
432C4#6393611yescharacteristic
433C4#6393711yescharacteristic
434C4D3×#21511yescharacteristic
435C4#6393911yescharacteristic
436C3—21no—
437C3#7112411yescharacteristic, complemented
438C3#7119911yescharacteristic, complemented
439C2—242no—
440C2—721no—
441C2#1059152011yescharacteristic
442C2#1059165111yescharacteristic
443C2#1059165911yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10593900C2
hypercenterC4⋊Q8D32
hypocenterC32#1642
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. #10592782C2semidirect
3. C3×#5094C2semidirect
4. C3×#5088C2semidirect
5. #10589090C2semidirect
6. #10589084C2semidirect
7. #10589082C2semidirect
31. #5094D3semidirect
32. #5088D3semidirect
128. C3×Dic6SD16semidirect
296. Dic6#215semidirect
377. C32#1642semidirect
437. C3#71124semidirect
438. C3#71199semidirect

Maximal subgroups

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

subgroupindex
#105939002
#105927822
C3×#50942
C3×#50882
#105890902
#105890842
#105890822
#711993
#711243