Finite Group Databasebeta

← #11778336

Subgroups — #11778336

Order 1152.

711 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 711 conjugacy classes of subgroups — too many to draw. Only its 88 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#10595981C211yesmaximal, characteristic
2C3×#4979C211yesmaximal, characteristic, complemented
3#10591174C212yesmaximal, complemented
4C3×#4528C211yesmaximal, characteristic, complemented
5#10589151C212yesmaximal, complemented
6#69706—31nomaximal
7#68133—31nomaximal
8C2×#3828—21no—
9C22×C3×Dic6C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11#64139—21no—
12#64139C2212yes—
13#64106—22no—
14#64104—22no—
15C2×C3×Dic3⋊C4—21no—
16C2×#3717—22no—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic
19C3×#830C2212yes—
20#63566—22no—
21#5510—31no—
22#5510—31no—
23#5510—61no—
24C3×#395—31no—
25#5227—32no—
26#4979D311yescharacteristic, complemented
27#4738—32no—
28C3×#324—31no—
29#4528D311yescharacteristic, complemented
30#4470—32no—
31#4451—32no—
32C22×C32×C4D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C2×#638—22no—
36#3828—42no—
37#3828—42no—
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)D412yes—
41C2×C4×C3⋊D3—22no—
42C2×#634—21no—
43C22×C3×Dic3—21no—
44C22×C3×Dic3—21no—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—42no—
47C32×C4⋊C4—22no—
48#3752—22no—
49#3752—22no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)D412yes—
52C3×Dic3⋊C4—42no—
53C2×C3×C3⋊C8—22no—
54#3717—44no—
55#1604—91noSylow
56C22×C3×Q8—31no—
57C2×#235—62no—
58C2×#235—61no—
59C2×#235—61no—
60C22×Dic6C2×D311yescharacteristic
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—61no—
63C2×C3×C4⋊C4C2×D311yescharacteristic
64C2×C3×C4⋊C4C2×D311yescharacteristic
65C2×#215—62no—
66#930—32no—
67#930—32no—
68#930—62no—
69#930—62no—
70#930—61no—
71#930—61no—
72C2×Dic3⋊C4—21no—
73C2×#204—62no—
74#897—62no—
75#897—62no—
76#897—64no—
77#895—62no—
78#895—62no—
79#895—64no—
80C3×C22⋊C8—32no—
81C3×#95—31no—
82C3×#95—61no—
83#835—62no—
84C3⋊#95—21no—
85C3⋊#95C2×D311yescharacteristic
86#830C2×D312yes—
87#816—62no—
88C23×C32C2×D411yescharacteristic
89C22×C3⋊D3—22no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4SD1612yescomplemented
96C2×C32×C4C2×D412yes—
97#638—42no—
98#638—42no—
99C2×C3⋊Dic3—22no—
100C4⋊(C3⋊D3)—42no—
101C4⋊(C3⋊D3)—42no—
102C4⋊(C3⋊D3)—81no—
103C4×C3⋊D3—44no—
104#634—42no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—41no—
110C2×C3×Dic3—41no—
111C3×Dic6—24no—
112C3×Dic6—42no—
113C3×C3⋊C8—42no—
114#537—91no—
115#448—92no—
116#395—31no—
117#327—92no—
118#324—31no—
119C2×C3×Q8—62no—
120C2×C3×Q8—122no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—61no—
125C22×C3×C4—61no—
126C22×C3×C4D1211yescharacteristic
127C22×C3×C4D1211yescharacteristic
128C22×C3×C4#6711yescharacteristic
129C22×C3×C4#6711yescharacteristic
130C22×C3×C4C22×D311yescharacteristic
131C22×C3×C4C22×D311yescharacteristic
132C2×#67—64no—
133C2×#67—62no—
134C2×#67—62no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137#235—124no—
138#235—122no—
139#235—122no—
140#235—124no—
141#235—122no—
142#235—122no—
143C2×D12—32no—
144C2×D12—32no—
145C2×D12—62no—
146C2×D12—62no—
147C2×D12—61no—
148C2×D12—61no—
149C2×D12—62no—
150C2×D12—61no—
151C2×D12—61no—
152C2×C4×D3—64no—
153C2×C4×D3—62no—
154C2×C4×D3—62no—
155C2×Dic6—22no—
156C2×Dic6—42no—
157C2×Dic6—62no—
158C2×Dic6—61no—
159C2×Dic6—61no—
160C2×C3×C8—62no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—122no—
165#215—124no—
166(C2×D3)⋊C4—64no—
167(C2×D3)⋊C4—62no—
168(C2×D3)⋊C4—62no—
169(C2×D3)⋊C4—64no—
170(C2×D3)⋊C4—62no—
171(C2×D3)⋊C4—62no—
172C4⋊Dic3—32no—
173C4⋊Dic3—32no—
174C4⋊Dic3—62no—
175C4⋊Dic3—62no—
176C4⋊Dic3—61no—
177C4⋊Dic3—61no—
178Dic3⋊C4—42no—
179C2×C3⋊C8—22no—
180#204—124no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32#12011yescharacteristic
184C22×C32C2×SD1611yescharacteristic
185C22×C32AGL(1,ℤ/8)11yescharacteristic
186C2×C3⋊D3—42no—
187C2×C3⋊D3—42no—
188C2×C3⋊D3—42no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—41no—
192C3⋊Dic3—42no—
193C3×Dic3—42no—
194C3×Dic3—42no—
195C2×#41—181no—
196C22×Q8—31no—
197C2×SD16—182no—
198C22⋊Q8—182no—
199#121—92no—
200#121—182no—
201#121—181no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4D3211yescharacteristic
204#103—182no—
205C22⋊C8—32no—
206#95—31no—
207#95—61no—
208C23×C3—21no—
209C23×C3C2×D1211yescharacteristic
210C23×C3C2×#6711yescharacteristic
211C22×D3—64no—
212C22×D3—62no—
213C22×D3—62no—
214C3×Q8—64no—
215C3×Q8—122no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4—121no—
236C2×C3×C4—121no—
237C2×C3×C4—121no—
238C2×C3×C4—121no—
239C2×C3×C4#20412yescomplemented
240C2×C3×C4#21512yescomplemented
241C2×C3×C4D3×D412yes—
242C2×C3×C4D3×D412yes—
243#67—124no—
244#67—124no—
245#67—122no—
246#67—122no—
247#67—122no—
248#67—122no—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—64no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258D12—122no—
259D12—122no—
260D12—124no—
261D12—122no—
262D12—122no—
263D12—242no—
264D12—242no—
265D12—241no—
266D12—241no—
267C4×D3—128no—
268C4×D3—124no—
269C4×D3—124no—
270Dic6—24no—
271Dic6—42no—
272Dic6—124no—
273Dic6—122no—
274Dic6—122no—
275C3×C8—122no—
276C3⋊C8—42no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32#45011yescharacteristic
280C2×C32#45611yescharacteristic
281C2×C32B(4,𝔽2)11yescharacteristic
282C3⋊D3—82no—
283#41—362no—
284#41—362no—
285C2×Q8—62no—
286C2×Q8—122no—
287C2×Q8—181no—
288C2×D4—92no—
289C2×D4—181no—
290C2×D4—181no—
291C2×D4—182no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—182no—
295C22×C4#62611yescharacteristic
296C22×C4#62611yescharacteristic
297C22×C4C2×D3211yescharacteristic
298SD16—364no—
299C2×C8—62no—
300C4⋊C4—22no—
301C4⋊C4—92no—
302C4⋊C4—122no—
303C4⋊C4—181no—
304C22⋊C4—182no—
305C22⋊C4—182no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3#88211yescharacteristic
316C22×C3C2×#20411yescharacteristic
317C22×C3#90811yescharacteristic
318C22×C3#93711yescharacteristic
319C22×C3C2×#21511yescharacteristic
320C22×C3#94911yescharacteristic
321C2×D3—124no—
322C2×D3—124no—
323C2×D3—124no—
324C2×D3—122no—
325C2×D3—122no—
326C2×D3—122no—
327C2×D3—122no—
328C2×D3—122no—
329C2×D3—122no—
330C3×C4—24no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—122no—
340C3×C4—122no—
341Dic3—42no—
342Dic3—42no—
343Dic3—124no—
344Dic3—122no—
345Dic3—122no—
346C32#160411yescharacteristic, complemented, Sylow
347C23—182no—
348C23C2×#62611yescharacteristic
349Q8—64no—
350Q8—122no—
351Q8—362no—
352D4—362no—
353D4—362no—
354D4—362no—
355D4—362no—
356D4—721no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—182no—
369C2×C4—364no—
370C2×C4#371712yescomplemented
371C2×C4#381112yes—
372C8—122no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3#473111yescharacteristic
383C2×C3#474111yescharacteristic
384C2×C3#475211yescharacteristic
385C2×C3#503211yescharacteristic
386C2×C3#504611yescharacteristic
387C2×C3#509511yescharacteristic
388D3—244no—
389D3—242no—
390D3—242no—
391C22—21no—
392C22—21no—
393C22—362no—
394C22—362no—
395C22—362no—
396C22#6382511yescharacteristic
397C22C2×#371711yescharacteristic
398C22#6398111yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—122no—
403C4—122no—
404C4—362no—
405C3—21no—
406C3#6813311yescharacteristic, complemented
407C3#6970611yescharacteristic, complemented
408C2—21no—
409C2—21no—
410C2—722no—
411C2#1059118311yescharacteristic
412C2#1059153211yescharacteristic
413C2#1059165011yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32#120
prime radical#10595981C2
hypercenterC2×C4⋊C4D32
hypocenterC32#1604
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4979C2semidirect
3. #10591174C2semidirect
4. C3×#4528C2semidirect
5. #10589151C2semidirect
26. #4979D3semidirect
29. #4528D3semidirect
95. C2×C32×C4SD16semidirect
239. C2×C3×C4#204semidirect
240. C2×C3×C4#215semidirect
346. C32#1604semidirect
370. C2×C4#3717semidirect
406. C3#68133semidirect
407. C3#69706semidirect

Maximal subgroups

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

subgroupindex
#105959812
C3×#49792
#105911742
C3×#45282
#105891512
#697063
#681333