Finite Group Databasebeta

← #11869024

Subgroups — #11869024

Order 1152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596283C211yesmaximal, characteristic
2#10595971C211yesmaximal, characteristic
3C2×C32×#103C211yesmaximal, characteristic
4#10592924C211yesmaximal, characteristic
5#10592805C211yesmaximal, characteristic
6#10592801C211yesmaximal, characteristic
7#10592663C211yesmaximal, characteristic
8#10592498C212yesmaximal, complemented
9#10592495C212yesmaximal, complemented
10#10592492C212yesmaximal, complemented
11#10592490C212yesmaximal, complemented
12#78779—34nomaximal
13C22×C32×Q8C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16Q8×C3⋊Dic3—21no—
17Q8×C3⋊Dic3—21no—
18Q8×C3⋊Dic3C2212yes—
19Q8×C3⋊Dic3C2212yes—
20#64135—22no—
21#64135C2211yescharacteristic
22#64135C2211yescharacteristic
23#64135C2211yescharacteristic
24#64133—22no—
25C2×#3748C2211yescharacteristic
26#64099—21no—
27#64099C2212yes—
28#64097—21no—
29#64097C2212yes—
30C32×#103C2212yes—
31C32×#103C2212yes—
32#63662C2212yes—
33#63662C2212yes—
34#63645C2211yescharacteristic
35#63645C2211yescharacteristic
36#63644C2211yescharacteristic
37#63644C2211yescharacteristic
38#63641C2212yes—
39#63641C2212yes—
40#63635C2212yes—
41#63634C2212yes—
42#5812—34no—
43#5500—34no—
44C2×C3×#103D314yes—
45#5225—34no—
46#5106—34no—
47#5102—34no—
48#4964—34no—
49#4799—38no—
50#4796—38no—
51#4793—38no—
52#4791—38no—
53C2×C32×Q8—21no—
54C2×C32×Q8—22no—
55C2×C32×Q8Q814yescomplemented
56C2×C32×Q8C2312yes—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4C2311yescharacteristic
60C2×#634—22no—
61C2×C32×C8—22no—
62C2×C32×C8C2311yescharacteristic
63C2×C32×C8C2311yescharacteristic
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2312yes—
66#3757—21no—
67#3757—21no—
68#3757D412yes—
69C4⋊(C3⋊Dic3)—22no—
70C4⋊(C3⋊Dic3)—22no—
71C4⋊(C3⋊Dic3)—22no—
72C4⋊(C3⋊Dic3)D411yescharacteristic
73C4⋊(C3⋊Dic3)D411yescharacteristic
74C4⋊(C3⋊Dic3)C2311yescharacteristic
75C4⋊(C3⋊Dic3)C2311yescharacteristic
76C4⋊(C3⋊Dic3)C2311yescharacteristic
77C4⋊(C3⋊Dic3)C2311yescharacteristic
78#3750—22no—
79#3750—22no—
80C4×C3⋊Dic3—21no—
81C4×C3⋊Dic3—21no—
82C4×C3⋊Dic3—22no—
83C4×C3⋊Dic3—22no—
84C4×C3⋊Dic3C2312yes—
85#3748—22no—
86C2×C3⋊(C3⋊C8)C2312yes—
87#3082—91noSylow
88C22×C3×Q8C2×D314yes—
89C22×C3×C8C2×D314yes—
90C2×C3×C4⋊C4C2×D314yes—
91Q8×Dic3—38no—
92Q8×Dic3—38no—
93Q8×Dic3—64no—
94Q8×Dic3—64no—
95#926—34no—
96#926—34no—
97#926—34no—
98#926—68no—
99#924—68no—
100C2×#208—34no—
101#890—38no—
102#890—64no—
103#888—38no—
104#888—64no—
105C3×#103C2×D318yes—
106C3×#103C2×D318yes—
107#835—38no—
108#835—38no—
109#818—34no—
110#818—34no—
111#817—34no—
112#817—34no—
113Dic3⋊C8—38no—
114Dic3⋊C8—38no—
115#808—38no—
116#807—38no—
117C23×C32C2×D411yescharacteristic
118C32×Q8—24no—
119C32×Q8—22no—
120C32×Q8C2×Q814yes—
121C2×C32×C4—24no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—81no—
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×Q812yes—
127C2×C32×C4#4112yes—
128C2×C32×C4C2411yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3C2×D412yes—
134C2×C3⋊Dic3C2×D412yes—
135#634—82no—
136C32×C8—21no—
137C32×C8—21no—
138C3⋊(C3⋊C8)—22no—
139#544—91no—
140#519—91no—
141#479—92no—
142#479—92no—
143#477—92no—
144#475—92no—
145#428—91no—
146#427—91no—
147#424—91no—
148#419—91no—
149C2×#103C3⋊D311yescharacteristic
150C2×C3×Q8—24no—
151C2×C3×Q8—28no—
152C2×C3×Q8Dic6116yescomplemented
153C2×C3×Q8C22×D318yes—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4C22×D314yes—
157C2×Dic6—68no—
158C2×C3×C8—28no—
159C2×C3×C8C22×D314yes—
160C2×C3×C8C22×D314yes—
161C3×C4⋊C4—24no—
162C3×C4⋊C4C22×D318yes—
163#217—38no—
164#217—64no—
165#217—64no—
166C4⋊Dic3—34no—
167C4⋊Dic3—34no—
168C4⋊Dic3—34no—
169C4⋊Dic3—34no—
170C4⋊Dic3—34no—
171C4⋊Dic3—34no—
172C4⋊Dic3—68no—
173C4⋊Dic3—68no—
174C4⋊Dic3—68no—
175Dic3⋊C4—68no—
176Dic3⋊C4—68no—
177C4×Dic3—38no—
178C4×Dic3—68no—
179C4×Dic3—68no—
180C4×Dic3—64no—
181C4×Dic3—64no—
182#208—68no—
183C2×C3⋊C8—38no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32C22⋊Q812yes—
187C22×C32C22×D411yescharacteristic
188C32×C4—24no—
189C32×C4—42no—
190C32×C4C22⋊Q812yes—
191C32×C4C22×Q811yescharacteristic
192C32×C4C2×#4111yescharacteristic
193C3⋊Dic3—22no—
194C3⋊Dic3—22no—
195C3⋊Dic3—41no—
196C3⋊Dic3—41no—
197C3⋊Dic3—41no—
198C3⋊Dic3—41no—
199C22×Q8C2×C3⋊D311yescharacteristic
200C2×M16—91no—
201C22×C8C2×C3⋊D311yescharacteristic
202C4⋊Q8—92no—
203C4⋊Q8—181no—
204#125—92no—
205#125—181no—
206C22⋊Q8—182no—
207C4×Q8—92no—
208C4×Q8—92no—
209C4×Q8—181no—
210C4×Q8—181no—
211#117—91no—
212#117—91no—
213#117—91no—
214#117—182no—
215C2×C4⋊C4C2×C3⋊D311yescharacteristic
216#107—91no—
217#107—91no—
218#107—92no—
219#106—91no—
220#106—91no—
221#106—92no—
222C4⋊C8—92no—
223C4⋊C8—92no—
224#103—92no—
225#103—92no—
226#103C2×C3⋊D312yes—
227#103C2×C3⋊D312yes—
228C23×C3D3×D414yes—
229C3×Q8—216no—
230C3×Q8—28no—
231C3×Q8C2×Dic6116yes—
232C2×C3×C4—216no—
233C2×C3×C4—28no—
234C2×C3×C4—28no—
235C2×C3×C4—84no—
236C2×C3×C4C2×Dic618yes—
237C2×C3×C4D3×D414yes—
238C2×C3×C4#23718yes—
239C2×C3×C4C23×D314yes—
240C2×Dic3—38no—
241C2×Dic3—38no—
242C2×Dic3—64no—
243C2×Dic3—64no—
244C2×Dic3—64no—
245C2×Dic3—64no—
246Dic6—248no—
247C3×C8—24no—
248C3×C8—24no—
249C3⋊C8—68no—
250C2×C32—22no—
251C2×C32C2×C22⋊Q811yescharacteristic
252C2×C32#57711yescharacteristic
253C2×C32#57811yescharacteristic
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—182no—
257C2×Q8#63414yescomplemented
258C2×Q8C22×C3⋊D312yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C22×C3⋊D311yescharacteristic
262M16—182no—
263C2×C8—22no—
264C2×C8—92no—
265C2×C8C22×C3⋊D311yescharacteristic
266C2×C8C22×C3⋊D311yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—91no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—91no—
272C4⋊C4—91no—
273C4⋊C4—91no—
274C4⋊C4—182no—
275C4⋊C4—182no—
276C4⋊C4—182no—
277C4⋊C4—182no—
278C4⋊C4—182no—
279C4⋊C4C22×C3⋊D312yes—
280C22⋊C4—92no—
281C22⋊C4—181no—
282C22⋊C4—181no—
283C42—92no—
284C42—182no—
285C42—182no—
286C42—181no—
287C42—181no—
288C22×C3—24no—
289C22×C3—24no—
290C22×C3#87818yes—
291C22×C3C2×D3×D414yes—
292C3×C4—216no—
293C3×C4—48no—
294C3×C4#87818yes—
295C3×C4C22×Dic614yes—
296C3×C4C2×#23714yes—
297Dic3—68no—
298Dic3—68no—
299Dic3—124no—
300Dic3—124no—
301Dic3—124no—
302Dic3—124no—
303C32#308211yescharacteristic, complemented, Sylow
304C23D4×C3⋊D311yescharacteristic
305Q8—24no—
306Q8—22no—
307Q8—722no—
308Q8C2×#63414yes—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—81no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4—181no—
319C2×C4C2×#63412yes—
320C2×C4D4×C3⋊D311yescharacteristic
321C2×C4#383012yes—
322C2×C4C23×C3⋊D311yescharacteristic
323C8—21no—
324C8—21no—
325C8—182no—
326C2×C3—28no—
327C2×C3C2×#87814yes—
328C2×C3#576214yes—
329C2×C3#576614yes—
330C22—21no—
331C22—21no—
332C22#6408712yes—
333C22C2×D4×C3⋊D311yescharacteristic
334C4—24no—
335C4—42no—
336C4—182no—
337C4—182no—
338C4—361no—
339C4—361no—
340C4—361no—
341C4—361no—
342C4#6408712yes—
343C4C22×#63411yescharacteristic
344C4C2×#383011yescharacteristic
345C3#7877914yescomplemented
346C2—22no—
347C2C2×#6408711yescharacteristic
348C2#1059623311yescharacteristic
349C2#1059623711yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×#103C2
socleC22×C32C22×D4
prime radicalC23×C32C2×D4
hypercenterC2×#103C3⋊D3
hypocenterC32#3082
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
8. #10592498C2semidirect
9. #10592495C2semidirect
10. #10592492C2semidirect
11. #10592490C2semidirect
55. C2×C32×Q8Q8semidirect
152. C2×C3×Q8Dic6semidirect
257. C2×Q8#634semidirect
303. C32#3082semidirect
345. C3#78779semidirect

Maximal subgroups

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

subgroupindex
#105962832
#105959712
C2×C32×#1032
#105929242
#105928052
#105928012
#105926632
#105924982
#105924952
#105924922
#105924902
#787793