Finite Group Databasebeta

← #11780101

Subgroups — #11780101

Order 1152.

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

normal subgroups only

This group has 529 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#10593895C211yesmaximal, characteristic
2#10592777C211yesmaximal, characteristic, complemented
3C3×#5087C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic, complemented
5#10589094C211yesmaximal, characteristic, complemented
6#10589080C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic, complemented
8#71110—31nomaximal
9#71048—31nomaximal
10C2×#3795—21no—
11C32×C4⋊Q8C2211yescharacteristic
12#64078C2211yescharacteristic
13C2×C3×#215—21no—
14C3×C4×D12C2211yescharacteristic
15C3×C4×D12C2211yescharacteristic
16#63908—21no—
17#63891—22no—
18D3×C4⋊Dic3—21no—
19#63857—21no—
20#63634—22no—
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#835—21no—
23C3×#809—21no—
24C3×#807—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4⋊(C3⋊C8)C2211yescharacteristic
27#5555—31no—
28#5551—31no—
29C3×#474—31no—
30C3×#461—31no—
31#5087D311yescharacteristic, complemented
32#5083D311yescharacteristic, complemented
33#5078—31no—
34#5078—31no—
35#5078—61no—
36D4⋊(C3⋊C8)—31no—
37D4⋊(C3⋊C8)—31no—
38#4486—31no—
39#4483—31no—
40#4483—31no—
41#4482—31no—
42C2×C32×Q8—21no—
43C2×#634—21no—
44C2×C3×D12D411yescharacteristic
45C2×C3×D12D411yescharacteristic
46C2×C3×C4×D3—21no—
47C2×C3×C4×D3—21no—
48C2×#625—22no—
49C2×D3×Dic3—22no—
50#3795—42no—
51#3795—42no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C42C2311yescharacteristic
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)D412yes—
57C2×C3⋊(C3⋊C8)—21no—
58C3×#215—42no—
59C3×(C2×D3)⋊C4—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×C4⋊Dic3D411yescharacteristic
62C3×C4⋊Dic3D411yescharacteristic
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65Dic3⋊Dic3—22no—
66(C2×D3)⋊Dic3—22no—
67#1647—91noSylow
68C2×#237—61no—
69C2×#235—61no—
70C2×C3×SD16—61no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4×D4—31no—
75C3×C4×D4—31no—
76C2×#215—21no—
77#938—61no—
78#930—61no—
79C2×C4⋊Dic3—61no—
80#924—61no—
81#897—61no—
82D3×C4⋊C4—61no—
83#886—62no—
84#883—62no—
85C4×D12C2×D311yescharacteristic
86C4×D12C2×D311yescharacteristic
87#869—31no—
88#869—31no—
89#869—61no—
90C3×#107—61no—
91C3×C4⋊C8—31no—
92C3×C4⋊C8—31no—
93C3×#103—61no—
94C3×#102—61no—
95#835—21no—
96#809—21no—
97#807—21no—
98#807—62no—
99#807—62no—
100#807—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—
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—22no—
115#634—41no—
116#634—41no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C3×D12—21no—
120C3×D12—21no—
121C3×D12—21no—
122C3×D12D812yes—
123C3×C4×D3—41no—
124C3×C4×D3—41no—
125C3×C4×D3—41no—
126C3×C4×D3—41no—
127#625—44no—
128D3×Dic3—44no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—41no—
131C3×C3⋊C8—41no—
132#547—91no—
133#500—91no—
134#474—31no—
135#461—31no—
136#332—91no—
137D4⋊C8—91no—
138D4⋊C8—91no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—41no—
142C2×C3×D4—31no—
143C2×C3×D4—31no—
144C22×C3×C4—61no—
145C22×C3×C4—61no—
146C2×#67—62no—
147C2×#67—62no—
148C22×Dic3—62no—
149#237—122no—
150#237—122no—
151#235—122no—
152#235—122no—
153C2×D12#6711yescharacteristic
154C2×D12#6711yescharacteristic
155C2×C4×D3—21no—
156C2×C4×D3—21no—
157C2×C4×D3—62no—
158C2×Dic6—61no—
159C2×Dic6—61no—
160C2×Dic6—121no—
161C3×SD16—122no—
162C2×C3×C8—61no—
163C2×C3×C8—61no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—41no—
172C3×C22⋊C4—61no—
173C3×C22⋊C4—61no—
174C3×C42—21no—
175C3×C42C22×D311yescharacteristic
176C3×C42C22×D311yescharacteristic
177#217—62no—
178#215—42no—
179(C2×D3)⋊C4—21no—
180(C2×D3)⋊C4—21no—
181(C2×D3)⋊C4—62no—
182C4⋊Dic3—32no—
183C4⋊Dic3—32no—
184C4⋊Dic3—62no—
185C4⋊Dic3—62no—
186C4⋊Dic3—61no—
187C4⋊Dic3—61no—
188C4⋊Dic3—121no—
189C4⋊Dic3#6711yescharacteristic
190C4⋊Dic3#6711yescharacteristic
191Dic3⋊C4—62no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—62no—
195C2×C3⋊C8—61no—
196C2×C3⋊C8—61no—
197C22×C32#12011yescharacteristic
198C2×C3×D3—22no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203C2×C3×D3—41no—
204C32×C4—21no—
205C32×C4—81no—
206C32×C4C2×D811yescharacteristic
207C32×C4#13511yescharacteristic
208C32×C4AGL(1,ℤ/8)11yescharacteristic
209C32×C4#13711yescharacteristic
210C3⋊Dic3—42no—
211C3×Dic3—41no—
212C3×Dic3—41no—
213C2×#41—181no—
214C2×SD16—61no—
215C4⋊Q8—91no—
216C4⋊Q8D3211yescharacteristic
217#123—182no—
218C22⋊Q8—181no—
219#121—181no—
220C4×D4—31no—
221C4×D4—31no—
222C2×C4⋊C4—181no—
223#107—61no—
224#107—182no—
225C4⋊C8—31no—
226C4⋊C8—31no—
227#103—61no—
228#102—61no—
229C4×C8—91no—
230C23×C3—61no—
231C23×C3—61no—
232C22×D3—21no—
233C22×D3—21no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—81no—
239C3×Q8—81no—
240C3×D4—32no—
241C3×D4—61no—
242C3×D4—61no—
243C3×D4—61no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—121no—
253C2×C3×C4—121no—
254C2×C3×C4—121no—
255C2×C3×C4—121no—
256C2×C3×C4D3×D411yescharacteristic
257C2×C3×C4D3×D411yescharacteristic
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4D3×D411yescharacteristic
260C2×C3×C4C2×#6711yescharacteristic
261C2×C3×C4C2×#6711yescharacteristic
262#67—124no—
263#67—124no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—122no—
269C2×Dic3—124no—
270D12—21no—
271D12—21no—
272D12—21no—
273D12#21312yes—
274C4×D3—41no—
275C4×D3—41no—
276C4×D3—41no—
277C4×D3—41no—
278C4×D3—124no—
279Dic6—121no—
280Dic6—121no—
281Dic6—121no—
282Dic6—121no—
283Dic6—241no—
284Dic6—241no—
285C3×C8—121no—
286C3×C8—121no—
287C3⋊C8—41no—
288C3⋊C8—41no—
289C3⋊C8—64no—
290C3⋊C8—62no—
291C3⋊C8—62no—
292C2×C32#44711yescharacteristic
293C2×C32#45211yescharacteristic
294C2×C32#45611yescharacteristic
295C3×D3—42no—
296C3×D3—81no—
297#41—362no—
298#41—362no—
299C2×Q8—21no—
300C2×Q8—181no—
301C2×D4—31no—
302C2×D4—31no—
303C2×D4—182no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—182no—
307SD16—122no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—181no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—31no—
314C4⋊C4—31no—
315C4⋊C4—92no—
316C4⋊C4—182no—
317C4⋊C4—181no—
318C22⋊C4—61no—
319C22⋊C4—61no—
320C22⋊C4—182no—
321C42C2×D3211yescharacteristic
322C22×C3—21no—
323C22×C3—62no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3#93711yescharacteristic
330C22×C3#93711yescharacteristic
331C2×D3—22no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—82no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4—121no—
348C3×C4—121no—
349C3×C4D3×D811yescharacteristic
350C3×C4#91111yescharacteristic
351C3×C4#91511yescharacteristic
352C3×C4#91611yescharacteristic
353C3×C4C2×#21311yescharacteristic
354C3×C4#94211yescharacteristic
355C3×C4#94911yescharacteristic
356C3×C4#95011yescharacteristic
357Dic3—41no—
358Dic3—41no—
359Dic3—122no—
360Dic3—122no—
361Dic3—242no—
362C32#164711yescharacteristic, complemented, Sylow
363C23—61no—
364C23—61no—
365Q8—41no—
366Q8—41no—
367Q8—361no—
368Q8—361no—
369D4—32no—
370D4—61no—
371D4—61no—
372D4—61no—
373D4—364no—
374C2×C4—41no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—182no—
382C2×C4—364no—
383C2×C4D3×#6711yescharacteristic
384C2×C4D3×#6711yescharacteristic
385C2×C4#381111yescharacteristic
386C8—121no—
387C8—121no—
388C8—182no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—122no—
393C2×C3—241no—
394C2×C3#503611yescharacteristic
395C2×C3#504711yescharacteristic
396C2×C3#515611yescharacteristic
397C2×C3#517311yescharacteristic
398C2×C3#520511yescharacteristic
399C2×C3#520511yescharacteristic
400D3—42no—
401D3—81no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22#6398311yescharacteristic
409C4—21no—
410C4—81no—
411C4—121no—
412C4—121no—
413C4—362no—
414C4D3×#21311yescharacteristic
415C4#6392511yescharacteristic
416C4#6394611yescharacteristic
417C4#6394711yescharacteristic
418C3—21no—
419C3#7104811yescharacteristic, complemented
420C3#7111011yescharacteristic, complemented
421C2—122no—
422C2—241no—
423C2#1059151511yescharacteristic
424C2#1059165511yescharacteristic
425C2#1059166011yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10593895C2
hypercenterC4⋊Q8D32
hypocenterC32#1647
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. #10592777C2semidirect
3. C3×#5087C2semidirect
4. C3×#5083C2semidirect
5. #10589094C2semidirect
6. #10589080C2semidirect
7. #10589080C2semidirect
31. #5087D3semidirect
32. #5083D3semidirect
362. C32#1647semidirect
419. C3#71048semidirect
420. C3#71110semidirect

Maximal subgroups

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

subgroupindex
#105938952
#105927772
C3×#50872
C3×#50832
#105890942
#105890802
#711103
#710483