Finite Group Databasebeta

← #11860911

Subgroups — #11860911

Order 1152.

778 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#64237C211yesmaximal, characteristic
2#10594051C212yesmaximal, complemented
3#10593738C211yesmaximal, characteristic, complemented
4C2×#63743C211yesmaximal, characteristic, complemented
5#10593496C212yesmaximal, complemented
6#10593445C211yesmaximal, characteristic, complemented
7C4×#3776C211yesmaximal, characteristic, complemented
8#10593432C212yesmaximal, complemented
9#10593431C212yesmaximal, complemented
10#10593420C211yesmaximal, characteristic, complemented
11C4×#3774C211yesmaximal, characteristic, complemented
12C2×#3805C2212yes—
13C2×#3792C2211yescharacteristic
14C22×#622C2211yescharacteristic
15C2×#3776C2211yescharacteristic
16C2×#3775C2212yes—
17#64237C418yescomplemented
18C2×#3774C2211yescharacteristic
19#64086C2211yescharacteristic
20C3×#67⋊C4—22no—
21Dic3×#67—22no—
22C2×#3724C2212yes—
23C4×#627C2212yes—
24#63902C2212yes—
25C4×#625C2212yes—
26#63877—22no—
27D3×Dic3⋊C4—22no—
28#63838C2212yes—
29#63777C2211yescharacteristic
30#63777C2211yescharacteristic
31C4×#622C2211yescharacteristic
32C4×#622C2211yescharacteristic
33#63743C2211yescharacteristic
34#63743C2211yescharacteristic
35#63743C2212yes—
36#63740C2212yes—
37#63740C2212yes—
38#63739C2212yes—
39#63739C2212yes—
40#5546—62no—
41C22×C3⋊Dic3C2311yescharacteristic
42C2×C3×#67—22no—
43C22×C3×Dic3—22no—
44C2×C3×C4×D3—22no—
45C2×#627C2312yes—
46C2×#625C2312yes—
47#3805—24no—
48#3805C2×C418yescomplemented
49C2×D3×Dic3—22no—
50C2×D3×Dic3—22no—
51#3792—22no—
52C2×#622C2×C414yescomplemented
53C2×#622C2311yescharacteristic
54C2×#622C2311yescharacteristic
55C2×#622C2311yescharacteristic
56C2×#622C2311yescharacteristic
57#3776C2×C414yescomplemented
58#3775C2×C418yescomplemented
59#3774C2×C414yescomplemented
60C32×C22⋊C4D411yescharacteristic
61#3757D411yescharacteristic
62#3750D411yescharacteristic
63#3750D411yescharacteristic
64C4×C3⋊Dic3C2311yescharacteristic
65C4×C3⋊Dic3C2311yescharacteristic
66C3×(C2×D3)⋊C4—22no—
67C3×Dic3⋊C4—22no—
68C3×C4×Dic3—22no—
69#3724—22no—
70#3724C2312yes—
71#3724C2312yes—
72Dic3⋊Dic3—22no—
73(C2×D3)⋊Dic3—22no—
74Dic32—22no—
75#2951—91nomaximal, Sylow
76C2×#237—62no—
77C3×C4×D4—62no—
78D4×Dic3—62no—
79C4×#67—62no—
80C4×#67—62no—
81C2×Dic3⋊C4—62no—
82C2×Dic3⋊C4—62no—
83D3×C4⋊C4—62no—
84#887—62no—
85#67⋊C4—22no—
86#67⋊C4—62no—
87#877—62no—
88#877—62no—
89#872—62no—
90C4×Dic6—62no—
91C23×C32C2×D411yescharacteristic
92C22×C3×D3—22no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C3⋊Dic3C22×C412yes—
96C2×C3⋊Dic3C2×D411yescharacteristic
97C2×C3⋊Dic3C2×D411yescharacteristic
98C2×C3⋊Dic3C2×D411yescharacteristic
99C2×C3⋊Dic3#4112yes—
100C2×C3⋊Dic3C2411yescharacteristic
101C3×#67—28no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—82no—
106C3×C4×D3—82no—
107#627—22no—
108#627C22×C414yes—
109#625—22no—
110#625C22×C414yes—
111D3×Dic3—28no—
112D3×Dic3—82no—
113#622—21no—
114#622—21no—
115#622C22×C412yes—
116#622C22×C412yes—
117C2×#135—91no—
118#520—92no—
119#440—92no—
120#440—92no—
121C4×Q16—91no—
122C4×Q16—91no—
123C4×D8—91no—
124C4×D8—91no—
125C2×#107—91no—
126#417—92no—
127#405—91no—
128C2×C3×D4—62no—
129C22×C3×C4—62no—
130C22×C3×C4—62no—
131C2×#67—22no—
132C2×#67—62no—
133C22×Dic3—22no—
134C22×Dic3—62no—
135C22×Dic3—62no—
136C22×Dic3—62no—
137#237—68no—
138#237—68no—
139C2×C4×D3—22no—
140C2×C4×D3—62no—
141C2×C4×D3—62no—
142C2×Dic6—62no—
143C3×C4⋊C4—62no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—62no—
146C3×C42—62no—
147#217—62no—
148#217—62no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—62no—
151C4⋊Dic3—62no—
152Dic3⋊C4—22no—
153Dic3⋊C4—62no—
154Dic3⋊C4—62no—
155Dic3⋊C4—62no—
156Dic3⋊C4—62no—
157Dic3⋊C4—62no—
158Dic3⋊C4—62no—
159Dic3⋊C4—62no—
160C4×Dic3—22no—
161C4×Dic3—62no—
162C4×Dic3—62no—
163C4×Dic3—62no—
164C4×Dic3—62no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32C4×D412yes—
168C22×C32C22×D411yescharacteristic
169C2×C3×D3—24no—
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C32×C4—21no—
173C32×C4—21no—
174C3⋊Dic3—21no—
175C3⋊Dic3—21no—
176C3⋊Dic3C4×D412yes—
177C3⋊Dic3C23×C411yescharacteristic
178C3⋊Dic3C2×#4111yescharacteristic
179C3×Dic3—24no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C2×#41—92no—
183#135—98no—
184C2×Q16—91no—
185C2×SD16—92no—
186C2×D8—91no—
187C2×M16—91no—
188C22×C8—91no—
189C4×Q8—92no—
190C4×Q8—92no—
191C4×D4—62no—
192C4×D4—92no—
193C4×D4—92no—
194C4×D4—182no—
195#117—91no—
196#117—182no—
197C2×C4⋊C4—92no—
198C2×C4⋊C4—182no—
199#107—91no—
200#107—91no—
201#107—92no—
202#103—92no—
203#103—92no—
204#102—92no—
205#102—92no—
206#97—91no—
207#97—91no—
208C4×C8—91no—
209C4×C8—91no—
210C23×C3—22no—
211C23×C3—62no—
212C22×D3—22no—
213C3×D4—68no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—242no—
220C2×C3×C4—242no—
221#67—28no—
222#67—64no—
223#67—64no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—64no—
228C2×Dic3—62no—
229C2×Dic3—64no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—68no—
234C2×Dic3—82no—
235C2×Dic3—242no—
236C4×D3—68no—
237C4×D3—82no—
238C4×D3—242no—
239Dic6—64no—
240Dic6—64no—
241C2×C32—22no—
242C2×C32C2×C4×D411yescharacteristic
243C2×C32#57611yescharacteristic
244C2×C32#57811yescharacteristic
245C3×D3—44no—
246#41—98no—
247#41—184no—
248C2×Q8—92no—
249C2×D4—62no—
250C2×D4—92no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—91no—
254C22×C4—182no—
255C22×C4—182no—
256Q16—94no—
257SD16—98no—
258D8—94no—
259M16—182no—
260C2×C8—91no—
261C2×C8—91no—
262C2×C8—94no—
263C2×C8—91no—
264C2×C8—91no—
265C4⋊C4—62no—
266C4⋊C4—92no—
267C4⋊C4—92no—
268C4⋊C4—91no—
269C4⋊C4—91no—
270C4⋊C4—182no—
271C4⋊C4—182no—
272C22⋊C4—62no—
273C22⋊C4—91no—
274C22⋊C4—182no—
275C22⋊C4GO+(4,𝔽2)11yescharacteristic
276C42—62no—
277C42—91no—
278C42—91no—
279C42—182no—
280C22×C3—22no—
281C22×C3—24no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—64no—
285C22×C3—122no—
286C22×C3—122no—
287C2×D3—24no—
288C2×D3—42no—
289C2×D3—42no—
290C3×C4—42no—
291C3×C4—42no—
292C3×C4—64no—
293C3×C4—122no—
294C3×C4—122no—
295Dic3—24no—
296Dic3—42no—
297Dic3—42no—
298Dic3—62no—
299Dic3—62no—
300Dic3—64no—
301Dic3—122no—
302Dic3—122no—
303C32#295111yescharacteristic, complemented, Sylow
304C23—62no—
305C23C2×GO+(4,𝔽2)11yescharacteristic
306Q8—94no—
307Q8—182no—
308D4—68no—
309D4—94no—
310D4—182no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—91no—
315C2×C4—91no—
316C2×C4—92no—
317C2×C4—92no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—188no—
321C2×C4—242no—
322C2×C4—242no—
323C2×C4—722no—
324C2×C4C2×GO+(4,𝔽2)11yescharacteristic
325C2×C4C2×GO+(4,𝔽2)11yescharacteristic
326C8—92no—
327C8—92no—
328C8—181no—
329C8—181no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—44no—
334C2×C3—124no—
335D3—44no—
336C22—21no—
337C22—21no—
338C22—64no—
339C22—122no—
340C22—122no—
341C22C4×GO+(4,𝔽2)12yes—
342C22C22×GO+(4,𝔽2)11yescharacteristic
343C4—21no—
344C4—21no—
345C4—64no—
346C4—91no—
347C4—91no—
348C4—92no—
349C4—122no—
350C4—122no—
351C4—181no—
352C4—181no—
353C3—22no—
354C2—22no—
355C2—124no—
356C2C2×C4×GO+(4,𝔽2)11yescharacteristic
357C2#1059651411yescharacteristic
358C2#1059651611yescharacteristic
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutatorC3⋊Dic3C23×C4
FittingC32×C22⋊C4D4
socleC22×C32C22×D4
prime radicalC2×#64237C2
cosocleC2×C3⋊Dic3C24
hypercenterC22⋊C4GO+(4,𝔽2)
hypocenterC32#2951
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #10594051C2semidirect
3. #10593738C2semidirect
4. C2×#63743C2semidirect
5. #10593496C2semidirect
6. #10593445C2semidirect
7. C4×#3776C2semidirect
8. #10593432C2semidirect
9. #10593431C2semidirect
10. #10593420C2semidirect
11. C4×#3774C2semidirect
17. #64237C4semidirect
48. #3805C2×C4semidirect
52. C2×#622C2×C4semidirect
57. #3776C2×C4semidirect
58. #3775C2×C4semidirect
59. #3774C2×C4semidirect
303. C32#2951semidirect

Maximal subgroups

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

subgroupindex
C2×#642372
#105940512
#105937382
C2×#637432
#105934962
#105934452
C4×#37762
#105934322
#105934312
#105934202
C4×#37742
#29519