Finite Group Databasebeta

← #11779533

Subgroups — #11779533

Order 1152.

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

normal subgroups only

This group has 522 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#10593896C211yesmaximal, characteristic
2#10592783C211yesmaximal, characteristic, complemented
3C3×#5084C211yesmaximal, characteristic, complemented
4C3×#5079C211yesmaximal, characteristic, complemented
5#10589093C211yesmaximal, characteristic, complemented
6#10589082C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71228—31nomaximal
9#71146—31nomaximal
10C2×#3796—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×C4×D12C2211yescharacteristic
14C3×#869C2211yescharacteristic
15#63909—21no—
16#63890—22no—
17D3×C4⋊Dic3—21no—
18#63871—21no—
19#63636—21no—
20#63635—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#835—22no—
23C3×#809—21no—
24C3×#808—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5726—31no—
28#5555—31no—
29C3×#487—31no—
30C3×#476—31no—
31#5084—31no—
32#5084—31no—
33#5084—61no—
34#5084D311yescharacteristic, complemented
35#5079D311yescharacteristic, complemented
36#4547—31no—
37D4⋊(C3⋊C8)—31no—
38#4483—31no—
39#4482—31no—
40#4459—31no—
41#4459—31no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×C3×D12D411yescharacteristic
46C2×C3×C4×D3—21no—
47C2×C3×Dic6—21no—
48C2×#626—22no—
49C2×D3×Dic3—22no—
50#3796—42no—
51#3796—42no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C42C2311yescharacteristic
55#3752—21no—
56C4⋊(C3⋊Dic3)D411yescharacteristic
57C2×C3⋊(C3⋊C8)—21no—
58C3×(C2×D3)⋊C4—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3D411yescharacteristic
61C3×C4⋊Dic3D412yes—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8—21no—
64Dic3⋊Dic3—22no—
65#3722—22no—
66#1635—91noSylow
67C2×#239—61no—
68C2×#237—61no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—31no—
71C3×C4⋊Q8C2×D311yescharacteristic
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4×D4—31no—
74#946—61no—
75#930—61no—
76C2×C4⋊Dic3—61no—
77#897—61no—
78#895—61no—
79#894—62no—
80D3×C4⋊C4—61no—
81#886—62no—
82C4×D12—31no—
83C4×D12—31no—
84C4×D12—61no—
85C4×D12C2×D311yescharacteristic
86#869C2×D311yescharacteristic
87C3×#106—61no—
88C3×C4⋊C8—31no—
89C3×#103—62no—
90C3×#102—61no—
91C3×C4×C8—31no—
92#835—22no—
93#809—21no—
94#809—61no—
95#809—61no—
96#809—62no—
97#808—21no—
98#808—61no—
99#808—61no—
100#808—62no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—61no—
104C4⋊(C3⋊C8)C2×D311yescharacteristic
105C4×C3⋊C8C2×D311yescharacteristic
106C22×C3⋊D3—21no—
107C22×C3×D3—21no—
108C32×Q8—42no—
109C2×C32×C4—41no—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C3⋊Dic3—21no—
114C4⋊(C3⋊D3)—21no—
115C4⋊(C3⋊D3)—21no—
116C4×C3⋊D3—41no—
117C4×C3⋊D3—41no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—22no—
120C3×D12—21no—
121C3×D12SD1612yes—
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124C3×Dic6—41no—
125C3×Dic6—41no—
126#626—44no—
127D3×Dic3—44no—
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—41no—
130C3×C3⋊C8—41no—
131#547—91no—
132#487—31no—
133#476—31no—
134#476—91no—
135#332—91no—
136D4⋊C8—91no—
137D4⋊C8—91no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—41no—
141C2×C3×Q8—61no—
142C2×C3×D4—31no—
143C22×C3×C4—61no—
144C2×#67—62no—
145C22×Dic3—62no—
146#239—122no—
147#239—122no—
148#237—122no—
149#237—122no—
150C2×D12—31no—
151C2×D12—31no—
152C2×D12—61no—
153C2×D12—62no—
154C2×D12#6711yescharacteristic
155C2×C4×D3—21no—
156C2×C4×D3—62no—
157C2×C4×D3—61no—
158C2×C4×D3—61no—
159C2×C4×D3—121no—
160C2×Dic6—21no—
161C2×C3×C8—61no—
162C2×C3×C8—61no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—32no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—61no—
172C3×C22⋊C4—61no—
173C3×C42—21no—
174C3×C42C22×D311yescharacteristic
175C3×C42C22×D311yescharacteristic
176(C2×D3)⋊C4—21no—
177(C2×D3)⋊C4—62no—
178(C2×D3)⋊C4—62no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—121no—
182C4⋊Dic3—21no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—61no—
186C4⋊Dic3—62no—
187C4⋊Dic3#6711yescharacteristic
188C4⋊Dic3#6712yes—
189Dic3⋊C4—62no—
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—41no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3×D3—22no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C32×C4—21no—
203C32×C4—81no—
204C32×C4C2×SD1611yescharacteristic
205C32×C4#13511yescharacteristic
206C32×C4AGL(1,ℤ/8)11yescharacteristic
207C32×C4#13711yescharacteristic
208C3⋊Dic3—41no—
209C3×Dic3—42no—
210C3×Dic3—41no—
211C2×#41—181no—
212C4⋊Q8—31no—
213C4⋊Q8D3211yescharacteristic
214#123—182no—
215C22⋊Q8—181no—
216#121—181no—
217C4×D4—31no—
218C4×D4—91no—
219C2×C4⋊C4—181no—
220#106—61no—
221#106—181no—
222C4⋊C8—31no—
223C4⋊C8—91no—
224#103—62no—
225#102—61no—
226#102—181no—
227C4×C8—31no—
228C23×C3—61no—
229C22×D3—21no—
230C22×D3—61no—
231C22×D3—61no—
232C22×D3—121no—
233C3×Q8—42no—
234C3×Q8—42no—
235C3×Q8—82no—
236C3×Q8—121no—
237C3×Q8—121no—
238C3×D4—32no—
239C3×D4—61no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—62no—
247C2×C3×C4—61no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4D3×D411yescharacteristic
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4D3×D411yescharacteristic
254C2×C3×C4C2×#6711yescharacteristic
255C2×C3×C4C2×#6711yescharacteristic
256#67—124no—
257C2×Dic3—21no—
258C2×Dic3—22no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—121no—
262C2×Dic3—124no—
263D12—21no—
264D12—62no—
265D12—61no—
266D12—61no—
267D12—61no—
268D12—61no—
269D12—121no—
270D12—124no—
271D12#21512yes—
272C4×D3—41no—
273C4×D3—41no—
274C4×D3—124no—
275C4×D3—121no—
276C4×D3—121no—
277C4×D3—121no—
278C4×D3—121no—
279C4×D3—241no—
280C4×D3—241no—
281Dic6—41no—
282Dic6—41no—
283C3×C8—121no—
284C3×C8—121no—
285C3⋊C8—41no—
286C3⋊C8—41no—
287C3⋊C8—122no—
288C3⋊C8—121no—
289C3⋊C8—121no—
290C2×C32#45011yescharacteristic
291C2×C32#45211yescharacteristic
292C2×C32#45411yescharacteristic
293C3⋊D3—81no—
294C3×D3—42no—
295#41—362no—
296#41—362no—
297C2×Q8—21no—
298C2×Q8—61no—
299C2×D4—31no—
300C2×D4—91no—
301C2×D4—182no—
302C22×C4—61no—
303C22×C4—182no—
304C22×C4—181no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—181no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—32no—
311C4⋊C4—31no—
312C4⋊C4—61no—
313C4⋊C4—91no—
314C4⋊C4—182no—
315C22⋊C4—61no—
316C22⋊C4—182no—
317C22⋊C4—181no—
318C42C2×D3211yescharacteristic
319C22×C3—21no—
320C22×C3—62no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3#93711yescharacteristic
324C22×C3#93711yescharacteristic
325C2×D3—22no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—122no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—241no—
336C2×D3—241no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—82no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4—121no—
348C3×C4—122no—
349C3×C4D3×SD1611yescharacteristic
350C3×C4#91411yescharacteristic
351C3×C4#91511yescharacteristic
352C3×C4#91611yescharacteristic
353C3×C4#91811yescharacteristic
354C3×C4#91911yescharacteristic
355C3×C4C2×#21511yescharacteristic
356C3×C4#94911yescharacteristic
357Dic3—42no—
358Dic3—41no—
359Dic3—121no—
360Dic3—121no—
361Dic3—241no—
362C32#163511yescharacteristic, complemented, Sylow
363C23—61no—
364C23—181no—
365Q8—42no—
366Q8—121no—
367Q8—121no—
368D4—32no—
369D4—61no—
370D4—181no—
371D4—181no—
372D4—364no—
373C2×C4—41no—
374C2×C4—61no—
375C2×C4—62no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—181no—
379C2×C4—364no—
380C2×C4—361no—
381C2×C4—361no—
382C2×C4D3×#6711yescharacteristic
383C2×C4D3×#6711yescharacteristic
384C2×C4#381111yescharacteristic
385C8—121no—
386C8—121no—
387C8—361no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—122no—
392C2×C3#504611yescharacteristic
393C2×C3#506211yescharacteristic
394C2×C3#516911yescharacteristic
395C2×C3#517311yescharacteristic
396C2×C3#518711yescharacteristic
397C2×C3#520111yescharacteristic
398D3—42no—
399D3—242no—
400D3—241no—
401D3—241no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—361no—
406C22—361no—
407C22—361no—
408C22#6398311yescharacteristic
409C4—21no—
410C4—81no—
411C4—122no—
412C4—121no—
413C4—361no—
414C4D3×#21511yescharacteristic
415C4#6393911yescharacteristic
416C4#6395011yescharacteristic
417C4#6395111yescharacteristic
418C3—21no—
419C3#7114611yescharacteristic, complemented
420C3#7122811yescharacteristic, complemented
421C2—122no—
422C2—721no—
423C2#1059152511yescharacteristic
424C2#1059165111yescharacteristic
425C2#1059165611yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10593896C2
hypercenterC4⋊Q8D32
hypocenterC32#1635
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. #10592783C2semidirect
3. C3×#5084C2semidirect
4. C3×#5079C2semidirect
5. #10589093C2semidirect
6. #10589082C2semidirect
7. #10589081C2semidirect
34. #5084D3semidirect
35. #5079D3semidirect
362. C32#1635semidirect
419. C3#71146semidirect
420. C3#71228semidirect

Maximal subgroups

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

subgroupindex
#105938962
#105927832
C3×#50842
C3×#50792
#105890932
#105890822
#105890812
#712283
#711463