Finite Group Databasebeta

← #11788762

Subgroups — #11788762

Order 1152.

855 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 855 conjugacy classes of subgroups — too many to draw. Only its 124 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#10595996C211yesmaximal, characteristic
2#10594780C211yesmaximal, characteristic, complemented
3#10591232C211yesmaximal, characteristic, complemented
4#10591096C212yesmaximal, complemented
5C3×#4531C212yesmaximal, complemented
6#68713—32nomaximal
7C2×#3828C2211yescharacteristic
8C22×C4⋊(C3⋊D3)C2211yescharacteristic
9C32×#117C2211yescharacteristic
10#64104—22no—
11#64091—22no—
12#64084C412yescomplemented
13#64083C412yescomplemented
14C3×#926C2212yes—
15C2×C3×#208C2212yes—
16#63818—22no—
17#63817—24no—
18#63581—22no—
19#63564—24no—
20#5745—32no—
21#5525—32no—
22#5525—61no—
23#5131—32no—
24#5006—32no—
25#5001—32no—
26C3×H(3,ℤ/4)—32no—
27#4531D312yescomplemented
28C23×C3⋊D3—21no—
29C22×C32×C4C2311yescharacteristic
30C2×#638D412yes—
31#3828—22no—
32#3828—22no—
33C2×C4⋊(C3⋊D3)—22no—
34C2×C4⋊(C3⋊D3)—22no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C4⋊(C3⋊D3)—24no—
37C2×C4⋊(C3⋊D3)C2×C412yes—
38C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
39C2×C4×C3⋊D3D412yes—
40C2×#634C2×C411yescharacteristic
41C32×C4⋊C4—21no—
42C32×C22⋊C4—21no—
43C32×C42C2×C412yes—
44#3752—24no—
45C3×#217—22no—
46C3×C4⋊Dic3—22no—
47C3×C4×Dic3—22no—
48C3×#208—22no—
49C3×#208—22no—
50C2×C3×C3⋊C8D414yescomplemented
51#3712—24no—
52#3710—24no—
53#1910—91noSylow
54C2×#235—32no—
55C2×#235—61no—
56C22×D12—32no—
57C22×D12—61no—
58C2×C3×M16—32no—
59C3×#117—21no—
60C3×#117—32no—
61C3×#117C2×D312yes—
62#926C2×D312yes—
63C2×#208C2×D312yes—
64#907—64no—
65D3×M16—64no—
66#901—64no—
67#895—64no—
68#895—64no—
69#882—64no—
70#882—64no—
71#875—34no—
72#875—62no—
73#874—34no—
74#874—62no—
75#821—64no—
76#809—64no—
77#806—64no—
78C23×C32C2×D411yescharacteristic
79C22×C3⋊D3—22no—
80C22×C3⋊D3—42no—
81C22×C3⋊D3—82no—
82C22×C3⋊D3C22⋊C412yes—
83C2×C32×C4—22no—
84C2×C32×C4C22×C411yescharacteristic
85C2×C32×C4C2×D412yes—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4#4112yes—
88#638—24no—
89C2×C3⋊Dic3C22⋊C412yes—
90C4⋊(C3⋊D3)—22no—
91C4⋊(C3⋊D3)—24no—
92C4⋊(C3⋊D3)—41no—
93C4⋊(C3⋊D3)—41no—
94C4⋊(C3⋊D3)—84no—
95C4×C3⋊D3—22no—
96C4×C3⋊D3—22no—
97#634—22no—
98C2×C3×Dic3—42no—
99C3×C3⋊C8—24no—
100#568—91no—
101#535—91no—
102#418—92no—
103#410—91no—
104H(3,ℤ/4)—32no—
105C23×D3—62no—
106C23×D3—62no—
107C22×C3×C4—21no—
108C22×C3×C4C22×D312yes—
109C2×#67—34no—
110C2×#67—62no—
111#235—62no—
112#235—62no—
113#235—62no—
114#235—62no—
115#235—122no—
116#235—122no—
117C2×D12—32no—
118C2×D12—34no—
119C2×D12—61no—
120C2×D12—62no—
121C2×D12—62no—
122C2×D12—62no—
123C2×D12—62no—
124C2×D12—62no—
125C2×D12—62no—
126C2×D12—62no—
127C2×D12—68no—
128C2×D12—68no—
129C2×D12—122no—
130C2×D12—122no—
131C2×C4×D3—34no—
132C2×C4×D3—62no—
133C2×Dic6—32no—
134C2×Dic6—61no—
135C3×M16—62no—
136C3×M16—62no—
137C2×C3×C8—34no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—62no—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—62no—
144C3×C42—22no—
145C3×C42—62no—
146C3×C42C4×D314yes—
147#217—22no—
148(C2×D3)⋊C4—68no—
149(C2×D3)⋊C4—68no—
150C4⋊Dic3—22no—
151C4×Dic3—22no—
152#208—22no—
153#208—22no—
154C2×C3⋊C8D1214yescomplemented
155ΓL(1,𝔽25)—68no—
156D3×C8—68no—
157C22×C32—41no—
158C22×C32C2×C22⋊C411yescharacteristic
159C22×C32#12112yes—
160C2×C3⋊D3—22no—
161C2×C3⋊D3—42no—
162C2×C3⋊D3—44no—
163C2×C3⋊D3—42no—
164C2×C3⋊D3—84no—
165C32×C4—42no—
166C32×C4C4×D412yes—
167C32×C4#12011yescharacteristic
168C32×C4#12311yescharacteristic
169C3⋊Dic3—22no—
170C3×Dic3—82no—
171C2×#41—91no—
172C22×D4—91no—
173#131—184no—
174C2×M16—32no—
175C2×M16—182no—
176#127—92no—
177#124—92no—
178#121—182no—
179#120—182no—
180#117—32no—
181#117D3211yescharacteristic
182#102—184no—
183#99—182no—
184C23×C3—21no—
185C23×C3D3×D412yes—
186C22×D3—34no—
187C22×D3—62no—
188C22×D3—64no—
189C22×D3—64no—
190C22×D3—124no—
191C22×D3—122no—
192C22×D3—122no—
193C22×D3—244no—
194C22×D3—244no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—122no—
202C2×C3×C4C2×C4×D312yes—
203C2×C3×C4C2×D1212yes—
204C2×C3×C4#23512yes—
205C2×C3×C4D3×D412yes—
206C2×C3×C4D3×D412yes—
207C2×C3×C4#23912yes—
208#67—64no—
209#67—64no—
210#67—124no—
211C2×Dic3—34no—
212C2×Dic3—42no—
213C2×Dic3—62no—
214D12—62no—
215D12—64no—
216D12—62no—
217D12—64no—
218D12—124no—
219D12—122no—
220D12—122no—
221D12—122no—
222D12—122no—
223D12—122no—
224D12—248no—
225D12—248no—
226C4×D3—64no—
227C4×D3—64no—
228C4×D3—122no—
229C4×D3—122no—
230Dic6—62no—
231Dic6—62no—
232Dic6—122no—
233C3×C8—64no—
234C3⋊C8—24no—
235C2×C32—21no—
236C2×C32—22no—
237C2×C32#38611yescharacteristic
238C3⋊D3—42no—
239C3⋊D3—82no—
240C24—181no—
241#41—182no—
242#41—182no—
243C2×Q8—91no—
244C2×D4—92no—
245C2×D4—91no—
246C2×D4—92no—
247C2×D4—184no—
248C2×D4—182no—
249C2×D4—182no—
250C2×D4—181no—
251C22×C4—92no—
252C22×C4C2×D3211yescharacteristic
253M16—62no—
254M16—62no—
255M16—184no—
256C2×C8—34no—
257C2×C8—184no—
258C4⋊C4—21no—
259C4⋊C4—62no—
260C22⋊C4—21no—
261C22⋊C4—62no—
262C22⋊C4—184no—
263C42—62no—
264C42#62412yes—
265C22×C3—22no—
266C22×C3—21no—
267C22×C3—42no—
268C22×C3—42no—
269C22×C3D3×C22⋊C412yes—
270C22×C3#88412yes—
271C22×C3#89512yes—
272C2×D3—64no—
273C2×D3—122no—
274C2×D3—124no—
275C2×D3—124no—
276C2×D3—124no—
277C2×D3—128no—
278C2×D3—128no—
279C2×D3—124no—
280C2×D3—244no—
281C2×D3—244no—
282C2×D3—248no—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—44no—
287C3×C4—44no—
288C3×C4—242no—
289C3×C4C4×D1212yes—
290C3×C4#88212yes—
291C3×C4#89312yes—
292C3×C4#89412yes—
293Dic3—64no—
294Dic3—82no—
295Dic3—122no—
296C32#191011yescharacteristic, complemented, Sylow
297C23—92no—
298C23—182no—
299C23—362no—
300C23—722no—
301C23#381111yescharacteristic
302Q8—182no—
303D4—184no—
304D4—182no—
305D4—184no—
306D4—361no—
307D4—361no—
308D4—724no—
309C2×C4—22no—
310C2×C4—92no—
311C2×C4—122no—
312C2×C4—182no—
313C2×C4—182no—
314C2×C4#379912yes—
315C2×C4D3×D1212yes—
316C2×C4C2×#62411yescharacteristic
317C2×C4#381111yescharacteristic
318C8—64no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—42no—
325C2×C3#466512yes—
326D3—124no—
327D3—242no—
328D3—244no—
329D3—244no—
330C22—41no—
331C22—182no—
332C22—362no—
333C22—364no—
334C22—362no—
335C22—724no—
336C22#6391012yes—
337C22#6397411yescharacteristic
338C4—42no—
339C4—182no—
340C4—242no—
341C4#6389412yes—
342C4#6389711yescharacteristic
343C4#6392311yescharacteristic
344C3—21no—
345C3#6871312yescomplemented
346C2—21no—
347C2—22no—
348C2—362no—
349C2—722no—
350C2#1059138011yescharacteristic
subgroupG/H
centerC2#10591380
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC2×C32#386
prime radical#10595996C2
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1910
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #10594780C2semidirect
3. #10591232C2semidirect
4. #10591096C2semidirect
5. C3×#4531C2semidirect
12. #64084C4semidirect
13. #64083C4semidirect
27. #4531D3semidirect
50. C2×C3×C3⋊C8D4semidirect
154. C2×C3⋊C8D12semidirect
296. C32#1910semidirect
345. C3#68713semidirect

Maximal subgroups

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

subgroupindex
#105959962
#105947802
#105912322
#105910962
C3×#45312
#687133