Finite Group Databasebeta

← #11780261

Subgroups — #11780261

Order 1152.

441 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 86 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#10593865C211yesmaximal, characteristic
2C3×#4851C211yesmaximal, characteristic, complemented
3#10590379C211yesmaximal, characteristic
4#10590312C211yesmaximal, characteristic, complemented
5#10589349C211yesmaximal, characteristic
6C3×#4486C211yesmaximal, characteristic, complemented
7#10589083C211yesmaximal, characteristic, complemented
8#71126—31nomaximal
9#68437—31nomaximal
10C4×C4⋊(C3⋊D3)C2211yescharacteristic
11C3×#869C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14#63871—21no—
15#63864—21no—
16#63849—21no—
17#63848—22no—
18Dic3×Dic6—21no—
19C2×#3717—21no—
20C32×C4⋊C8C2211yescharacteristic
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#817—21no—
23C3×#816—21no—
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25#63576—22no—
26#63566—21no—
27#5732—31no—
28#5584—31no—
29C3×#475—31no—
30#5083—31no—
31#5078—31no—
32#5078—31no—
33#4851D311yescharacteristic, complemented
34#4841—31no—
35C3×#332—31no—
36#4535—31no—
37#4486D311yescharacteristic, complemented
38#4484—31no—
39#4483—31no—
40#4483—31no—
41#4483—61no—
42C2×C4⋊(C3⋊D3)D411yescharacteristic
43C2×C4×C3⋊D3—21no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6D411yescharacteristic
47C2×C32×C8—21no—
48C32×C42C2311yescharacteristic
49#3752—21no—
50C4⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3⋊(C3⋊C8)—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3D412yes—
56C3×C4⋊Dic3D411yescharacteristic
57C3×Dic3⋊C4—21no—
58C3×C4×Dic3—21no—
59C2×C3×C3⋊C8—21no—
60Dic3⋊Dic3—22no—
61#3722—22no—
62#3722—22no—
63Dic32—22no—
64#3717—42no—
65#1650—91noSylow
66C3×C4⋊Q8—31no—
67C3×C4⋊Q8—31no—
68C3×C4×Q8—31no—
69#947—61no—
70#946—61no—
71Q8×Dic3—61no—
72C2×#215—61no—
73C2×#204—61no—
74#898—62no—
75#898—62no—
76#897—61no—
77#892—61no—
78#892—61no—
79#875—61no—
80C4×D12—31no—
81C4×D12—31no—
82C4×D12—61no—
83#869C2×D311yescharacteristic
84#869C2×D311yescharacteristic
85C4×Dic6—61no—
86C4×Dic6C2×D311yescharacteristic
87C3×#106—61no—
88C3×C4⋊C8—21no—
89C3×C4⋊C8—31no—
90C3×C4⋊C8C2×D311yescharacteristic
91C3×C4⋊C8C2×D311yescharacteristic
92C3×#103—61no—
93#835—61no—
94#817—21no—
95#816—21no—
96#816—61no—
97#807—62no—
98#807—62no—
99C4⋊(C3⋊C8)C2×D311yescharacteristic
100C4×C3⋊C8—31no—
101C4×C3⋊C8—31no—
102C4×C3⋊C8—61no—
103C22×C3⋊D3—21no—
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C32×C4C2×D411yescharacteristic
107C2×C3⋊Dic3—21no—
108C4⋊(C3⋊D3)—21no—
109C4⋊(C3⋊D3)—21no—
110C4×C3⋊D3—41no—
111C4×C3⋊D3—41no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C3×Dic6—21no—
117C3×Dic6—21no—
118C3×Dic6—41no—
119C3×Dic6—41no—
120C3×Dic6—41no—
121C3×Dic6—41no—
122C32×C8—41no—
123C3⋊(C3⋊C8)—22no—
124C3×C3⋊C8—41no—
125#556—91no—
126#500—91no—
127#475—31no—
128#461—91no—
129#332—31no—
130Q8⋊C8—91no—
131D4⋊C8—91no—
132C2×C3×Q8—31no—
133C2×C3×Q8—61no—
134C2×C3×Q8—61no—
135C2×D12—31no—
136C2×D12—31no—
137C2×D12—61no—
138C2×C4×D3—61no—
139C2×C4×D3—61no—
140C2×C4×D3—121no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C2×Dic6#6711yescharacteristic
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—41no—
147C2×C3×C8—61no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—61no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—61no—
154C3×C42—21no—
155C3×C42—61no—
156C3×C42C22×D311yescharacteristic
157C3×C42C22×D311yescharacteristic
158#215—122no—
159(C2×D3)⋊C4—62no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—62no—
162(C2×D3)⋊C4—62no—
163(C2×D3)⋊C4—61no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4—121no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—61no—
172C4⋊Dic3—62no—
173C4⋊Dic3D1212yes—
174C4⋊Dic3#6711yescharacteristic
175Dic3⋊C4—21no—
176Dic3⋊C4—62no—
177C4×Dic3—21no—
178C4×Dic3—62no—
179C4×Dic3—62no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—61no—
183C2×C3⋊C8—121no—
184#204—122no—
185C22×C32#12011yescharacteristic
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C32×C4—21no—
190C32×C4#13511yescharacteristic
191C32×C4#13511yescharacteristic
192C32×C4AGL(1,ℤ/8)11yescharacteristic
193C32×C4#13711yescharacteristic
194C3⋊Dic3—41no—
195C3×Dic3—42no—
196C3×Dic3—41no—
197C3×Dic3—41no—
198C3×Dic3—81no—
199C2×SD16—181no—
200C4⋊Q8—31no—
201C4⋊Q8—31no—
202#126—182no—
203#124—181no—
204C22⋊Q8—181no—
205C4×Q8—31no—
206C4×Q8—181no—
207C4×D4—91no—
208#117—181no—
209#107—182no—
210#106—61no—
211C4⋊C8—31no—
212C4⋊C8D3211yescharacteristic
213#103—61no—
214#103—181no—
215C4×C8—91no—
216C22×D3—61no—
217C22×D3—61no—
218C22×D3—121no—
219C3×Q8—61no—
220C3×Q8—61no—
221C3×Q8—121no—
222C3×Q8—121no—
223C3×Q8—121no—
224C3×Q8—121no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—62no—
231C2×C3×C4—121no—
232C2×C3×C4C2×D1211yescharacteristic
233C2×C3×C4D3×D411yescharacteristic
234C2×C3×C4D3×D411yescharacteristic
235C2×C3×C4D3×D411yescharacteristic
236C2×C3×C4D3×D411yescharacteristic
237C2×C3×C4C2×#6711yescharacteristic
238C2×Dic3—22no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—41no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—121no—
245D12—62no—
246D12—61no—
247D12—61no—
248D12—61no—
249D12—61no—
250D12—121no—
251C4×D3—121no—
252C4×D3—121no—
253C4×D3—121no—
254C4×D3—121no—
255C4×D3—241no—
256C4×D3—241no—
257Dic6—21no—
258Dic6—21no—
259Dic6—41no—
260Dic6—41no—
261Dic6—41no—
262Dic6—41no—
263C3×C8—41no—
264C3×C8—41no—
265C3×C8—81no—
266C3×C8—121no—
267C3⋊C8—41no—
268C3⋊C8—62no—
269C3⋊C8—62no—
270C3⋊C8—122no—
271C2×C32#44911yescharacteristic
272C2×C32#45211yescharacteristic
273C2×C32#45611yescharacteristic
274C3⋊D3—81no—
275C2×Q8—31no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×D4—91no—
279C22×C4—181no—
280SD16—362no—
281C2×C8—21no—
282C2×C8—61no—
283C2×C8—181no—
284C4⋊C4—32no—
285C4⋊C4—31no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—91no—
291C4⋊C4—182no—
292C22⋊C4—182no—
293C22⋊C4—182no—
294C22⋊C4—181no—
295C42—61no—
296C42—182no—
297C42C2×D3211yescharacteristic
298C22×C3—21no—
299C22×C3#88211yescharacteristic
300C22×C3#93711yescharacteristic
301C2×D3—122no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—241no—
309C2×D3—241no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—41no—
317C3×C4—122no—
318C3×C4—121no—
319C3×C4—121no—
320C3×C4—241no—
321C3×C4#91111yescharacteristic
322C3×C4#91411yescharacteristic
323C3×C4#91611yescharacteristic
324C3×C4#91611yescharacteristic
325C3×C4#91611yescharacteristic
326C3×C4#91811yescharacteristic
327C3×C4#94211yescharacteristic
328C3×C4#95011yescharacteristic
329Dic3—41no—
330Dic3—42no—
331Dic3—41no—
332Dic3—81no—
333Dic3—121no—
334Dic3—121no—
335Dic3—241no—
336C32#165011yescharacteristic, complemented, Sylow
337C23—181no—
338Q8—61no—
339Q8—61no—
340Q8—121no—
341Q8—121no—
342Q8—121no—
343Q8—121no—
344D4—181no—
345D4—181no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—62no—
349C2×C4—121no—
350C2×C4—181no—
351C2×C4—361no—
352C2×C4—361no—
353C2×C4D3×D1211yescharacteristic
354C2×C4#380211yescharacteristic
355C2×C4D3×#6711yescharacteristic
356C8—41no—
357C8—121no—
358C8—182no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3#475211yescharacteristic
363C2×C3#478111yescharacteristic
364C2×C3#480811yescharacteristic
365C2×C3#504711yescharacteristic
366C2×C3#517211yescharacteristic
367C2×C3#520511yescharacteristic
368D3—242no—
369D3—241no—
370D3—241no—
371C22—361no—
372C22—361no—
373C22—361no—
374C22#6392211yescharacteristic
375C4—21no—
376C4—121no—
377C4—121no—
378C4—122no—
379C4—241no—
380C4—361no—
381C4#6379811yescharacteristic
382C4#6380911yescharacteristic
383C4#6393511yescharacteristic
384C4#6394811yescharacteristic
385C3—21no—
386C3#6843711yescharacteristic, complemented
387C3#7112611yescharacteristic, complemented
388C2—721no—
389C2#1059061611yescharacteristic
390C2#1059096411yescharacteristic
391C2#1059101811yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#120
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊C8D32
hypocenterC32#1650
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#4851C2semidirect
4. #10590312C2semidirect
6. C3×#4486C2semidirect
7. #10589083C2semidirect
33. #4851D3semidirect
37. #4486D3semidirect
336. C32#1650semidirect
386. C3#68437semidirect
387. C3#71126semidirect

Maximal subgroups

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

subgroupindex
#105938652
C3×#48512
#105903792
#105903122
#105893492
C3×#44862
#105890832
#711263
#684373