Finite Group Databasebeta

← #11778218

Subgroups — #11778218

Order 1152.

717 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 80 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#10595084C211yesmaximal, characteristic
2#10592930C211yesmaximal, characteristic, complemented
3C3×#5227C212yesmaximal, complemented
4#10589150C212yesmaximal, complemented
5#10589146C211yesmaximal, characteristic, complemented
6#70972—32nomaximal
7C22×C32×Q8C2211yescharacteristic, complemented
8C2×#3811—21no—
9C2×#3752—21no—
10C2×#3752C2211yescharacteristic
11C2×C3×#215—22no—
12C3×#930C2212yes—
13#63980—21no—
14#63922—22no—
15#63854—22no—
16#63658—21no—
17#63658C2211yescharacteristic, complemented
18C3×#835—22no—
19C3×#830C2212yescomplemented
20#5605—32no—
21C3×#448—32no—
22#5231—32no—
23#5231—61no—
24#5227D312yescomplemented
25#4537—32no—
26#4470—32no—
27#4468—32no—
28C23×C3⋊D3D411yescharacteristic
29C22×D32—21no—
30C2×C32×Q8—22no—
31C2×C32×Q8—42no—
32C22×C32×C4—21no—
33C22×C32×C4C2311yescharacteristic
34C22×C3⋊Dic3—21no—
35C22×C3⋊Dic3D411yescharacteristic
36C2×C3×#67—22no—
37C2×C3×D12D412yes—
38#3811—42no—
39#3811—42no—
40C2×#626—22no—
41C2×#625—21no—
42C2×#624—21no—
43#3752—22no—
44#3752—42no—
45C3×#215—44no—
46C3×(C2×D3)⋊C4—22no—
47C3×C4⋊Dic3D412yes—
48C2×C3×C3⋊C8—22no—
49#3724—21no—
50(C2×D3)⋊Dic3—22no—
51#1601—91noSylow
52C22×C3×Q8—21no—
53C22×C3×Q8C2×D312yescomplemented
54C2×D3×D4—62no—
55C2×C3×SD16—62no—
56C3×#121—32no—
57C2×#215—22no—
58#939—62no—
59#937—62no—
60#930C2×D312yes—
61C2×(C2×D3)⋊C4—32no—
62C2×(C2×D3)⋊C4—61no—
63C2×(C2×D3)⋊C4—62no—
64C2×(C2×D3)⋊C4—62no—
65#894—62no—
66#883—62no—
67#882—62no—
68C3×#103—62no—
69C3×C22⋊C8—32no—
70#835—22no—
71C3⋊#95—32no—
72C3⋊#95—61no—
73C3⋊#95—62no—
74C3⋊#95—62no—
75#830C2×D312yescomplemented
76C23×C32C2×D411yescharacteristic
77C22×C3⋊D3—21no—
78C22×C3⋊D3—21no—
79C22×C3⋊D3—41no—
80C22×C3⋊D3—41no—
81C22×C3⋊D3SD1612yescomplemented
82C22×C3×D3—22no—
83C2×D32—42no—
84C2×D32—81no—
85C2×D32—81no—
86C32×Q8—24no—
87C32×Q8—42no—
88C2×C32×C4—22no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4C2×D412yes—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—41no—
95C2×C3⋊Dic3—41no—
96C2×C3⋊Dic3—41no—
97C2×C3⋊Dic3—41no—
98C3×#67—42no—
99C3×#67—42no—
100C2×C3×Dic3—22no—
101C3×D12—42no—
102#626—44no—
103#625—81no—
104#624—42no—
105C3×C3⋊C8—42no—
106#534—91no—
107#448—32no—
108#393—91no—
109#327—92no—
110#323—91no—
111C23×D3—32no—
112C23×D3—61no—
113C23×D3—62no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—42no—
117C2×C3×Q8—44no—
118C2×C3×Q8—44no—
119C2×C3×D4—32no—
120C2×C3×D4—62no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4C22×D312yes—
125C2×#67—22no—
126C2×#67—62no—
127C2×#67—62no—
128C22×Dic3—32no—
129C22×Dic3—61no—
130C22×Dic3—62no—
131C22×Dic3—62no—
132D3×D4—124no—
133D3×D4—124no—
134C2×D12—62no—
135C2×D12#6712yes—
136C2×C4×D3—62no—
137C3×SD16—124no—
138C2×C3×C8—62no—
139C3×C4⋊C4—32no—
140C3×C22⋊C4—62no—
141#217—62no—
142#215—44no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—62no—
145(C2×D3)⋊C4—62no—
146(C2×D3)⋊C4—62no—
147(C2×D3)⋊C4—122no—
148(C2×D3)⋊C4—124no—
149(C2×D3)⋊C4—124no—
150C4⋊Dic3#6712yes—
151Dic3⋊C4—62no—
152C2×C3⋊C8—22no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32#12011yescharacteristic
156C22×C32C2×SD1611yescharacteristic
157C22×C32AGL(1,ℤ/8)11yescharacteristic
158C2×C3⋊D3—22no—
159C2×C3⋊D3—42no—
160C2×C3⋊D3—41no—
161C2×C3⋊D3—41no—
162C2×C3⋊D3—81no—
163C2×C3×D3—42no—
164C2×C3×D3—42no—
165C2×C3×D3—42no—
166D32—84no—
167C32×C4—22no—
168C32×C4—42no—
169C3⋊Dic3—41no—
170C3⋊Dic3—81no—
171C3×Dic3—42no—
172C22×Q8D3211yescharacteristic, complemented
173C22×D4—181no—
174C2×SD16—62no—
175#123—182no—
176#121—32no—
177#121—181no—
178#120—182no—
179C2×C22⋊C4—91no—
180C2×C22⋊C4—181no—
181#103—62no—
182C22⋊C8—32no—
183#95—91no—
184#95—181no—
185C23×C3—21no—
186C23×C3—62no—
187C23×C3D3×D412yes—
188C22×D3—22no—
189C22×D3—34no—
190C22×D3—62no—
191C22×D3—62no—
192C22×D3—62no—
193C22×D3—121no—
194C22×D3—121no—
195C22×D3—124no—
196C22×D3—122no—
197C22×D3—122no—
198C22×D3—122no—
199C22×D3—122no—
200C22×D3—242no—
201C22×D3—242no—
202C3×Q8—24no—
203C3×Q8—24no—
204C3×Q8—44no—
205C3×Q8—42no—
206C3×Q8—42no—
207C3×Q8—44no—
208C3×D4—122no—
209C3×D4—122no—
210C3×D4—122no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—22no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—62no—
221C2×C3×C4D3×D412yes—
222C2×C3×C4C2×#6712yes—
223#67—42no—
224#67—42no—
225#67—124no—
226#67—242no—
227C2×Dic3—22no—
228C2×Dic3—62no—
229C2×Dic3—121no—
230C2×Dic3—122no—
231C2×Dic3—122no—
232C2×Dic3—122no—
233C2×Dic3—122no—
234C2×Dic3—122no—
235C2×Dic3—122no—
236C2×Dic3—122no—
237C2×Dic3—122no—
238D12—42no—
239D12—124no—
240C4×D3—124no—
241C3×C8—122no—
242C3⋊C8—42no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32#45011yescharacteristic
246C2×C32#45511yescharacteristic
247C2×C32B(4,𝔽2)11yescharacteristic
248C3⋊D3—42no—
249C3×D3—82no—
250C24—91no—
251C24—181no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×D4—32no—
255C2×D4—62no—
256C2×D4—182no—
257C2×D4—181no—
258C2×D4—362no—
259C2×D4—362no—
260C22×C4—21no—
261C22×C4—91no—
262C22×C4—181no—
263C22×C4—181no—
264C22×C4C2×D3211yescharacteristic
265SD16—124no—
266C2×C8—62no—
267C4⋊C4—32no—
268C4⋊C4—181no—
269C22⋊C4—62no—
270C22⋊C4—182no—
271C22⋊C4—182no—
272C22⋊C4—362no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—122no—
281C22×C3—122no—
282C22×C3—122no—
283C22×C3D3×SD1612yes—
284C22×C3#91412yes—
285C22×C3#93712yes—
286C2×D3—42no—
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—64no—
290C2×D3—122no—
291C2×D3—122no—
292C2×D3—122no—
293C2×D3—124no—
294C2×D3—124no—
295C2×D3—241no—
296C2×D3—241no—
297C2×D3—248no—
298C2×D3—242no—
299C2×D3—242no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—42no—
303C3×C4—44no—
304C3×C4—44no—
305C3×C4—122no—
306Dic3—42no—
307Dic3—122no—
308Dic3—241no—
309Dic3—242no—
310Dic3—242no—
311C32#160111yescharacteristic, complemented, Sylow
312C23—62no—
313C23—92no—
314C23—181no—
315C23—181no—
316C23—361no—
317C23—361no—
318C23—362no—
319C23—721no—
320C23—721no—
321C23#381111yescharacteristic
322Q8—24no—
323Q8—42no—
324D4—122no—
325D4—122no—
326D4—122no—
327D4—364no—
328D4—721no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—62no—
334C2×C4—181no—
335C2×C4—362no—
336C2×C4—361no—
337C2×C4—361no—
338C2×C4—361no—
339C2×C4—361no—
340C2×C4D3×#6712yes—
341C8—122no—
342C2×C3—22no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—22no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—242no—
350C2×C3#503212yes—
351C2×C3#516912yes—
352C2×C3#520212yes—
353D3—82no—
354D3—124no—
355D3—242no—
356C22—21no—
357C22—21no—
358C22—122no—
359C22—122no—
360C22—122no—
361C22—182no—
362C22—362no—
363C22—361no—
364C22—361no—
365C22—721no—
366C22—724no—
367C22#6394011yescharacteristic
368C22#6394111yescharacteristic
369C22#6398311yescharacteristic
370C4—22no—
371C4—42no—
372C4—122no—
373C4—361no—
374C4—721no—
375C3—21no—
376C3#7097212yescomplemented
377C2—21no—
378C2—21no—
379C2—242no—
380C2—362no—
381C2#1059165211yescharacteristic
382C2#1059165811yescharacteristic
383C2#1059172511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radical#10595084C2
hypercenterC22×Q8D32
hypocenterC32#1601
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #10592930C2semidirect
3. C3×#5227C2semidirect
4. #10589150C2semidirect
5. #10589146C2semidirect
7. C22×C32×Q8C22semidirect
17. #63658C22semidirect
19. C3×#830C22semidirect
24. #5227D3semidirect
53. C22×C3×Q8C2×D3semidirect
75. #830C2×D3semidirect
81. C22×C3⋊D3SD16semidirect
172. C22×Q8D32semidirect
311. C32#1601semidirect
376. C3#70972semidirect

Maximal subgroups

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

subgroupindex
#105950842
#105929302
C3×#52272
#105891502
#105891462
#709723