Finite Group Databasebeta

← #11785724

Subgroups — #11785724

Order 1152.

727 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 126 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×#63900C211yesmaximal, characteristic
2#10592825C211yesmaximal, characteristic
3#10591074C211yesmaximal, characteristic
4C3×#4554C212yesmaximal, complemented
5#10589167C212yesmaximal, complemented
6#68990—32nomaximal
7C22×C4×C3⋊D3C2211yescharacteristic
8C22×#624C412yes—
9C2×C32×M16C2211yescharacteristic
10C2×#3748C2211yescharacteristic
11C2×C3×C4⋊Dic3C2212yes—
12C2×C3×#208C2212yes—
13C2×#3724C412yes—
14#63900—21no—
15#63900—21no—
16#63900—21no—
17#63900—21no—
18#63647—21no—
19#63647—21no—
20#63557—21no—
21#63557—21no—
22C2×D3×C4⋊C4—32no—
23#5126—32no—
24#5126—32no—
25#5126—61no—
26C3×#343—32no—
27#4554—32no—
28#4554D312yescomplemented
29#4530—32no—
30C23×C3⋊D3C2×C411yescharacteristic
31C22×C32×C4C2311yescharacteristic
32C22×C3⋊Dic3C2×C411yescharacteristic
33C2×C4×C3⋊D3—22no—
34C2×C4×C3⋊D3—22no—
35C2×C4×C3⋊D3D411yescharacteristic
36C2×C4×C3⋊D3D411yescharacteristic
37C2×C4×C3⋊D3D411yescharacteristic
38C2×C4×C3⋊D3D411yescharacteristic
39C22×C3×Dic3C2×C414yes—
40C2×#624—22no—
41C2×#624—22no—
42C2×#624—44no—
43C32×M16—22no—
44C2×C32×C8—21no—
45#3748—22no—
46C2×C3⋊(C3⋊C8)—21no—
47C3×C4⋊Dic3—22no—
48C3×C4⋊Dic3—22no—
49C3×#208—22no—
50C3×#208—22no—
51C2×C3×C3⋊C8—22no—
52#3724—22no—
53#3724—22no—
54#1821—91noSylow
55C22×C4×D3—32no—
56C22×C4×D3—34no—
57C22×C4×D3—61no—
58C2×C3×M16—21no—
59C2×C3×M16—32no—
60C2×C3×M16C2×D312yes—
61C2×C3×C4⋊C4—32no—
62C2×C4⋊Dic3C2×D312yes—
63C2×Dic3⋊C4—34no—
64C2×#208—32no—
65C2×#208—61no—
66C2×#208C2×D312yes—
67D3×C4⋊C4—62no—
68D3×C4⋊C4—62no—
69D3×C4⋊C4—62no—
70D3×C4⋊C4—62no—
71(C2×D3)⋊C8—62no—
72(C2×D3)⋊C8—62no—
73(C2×D3)⋊C8—62no—
74(C2×D3)⋊C8—62no—
75(C2×D3)⋊C8—62no—
76(C2×D3)⋊C8—62no—
77C23×C32C22×C411yescharacteristic
78C22×C3⋊D3—22no—
79C22×C3⋊D3—22no—
80C22×C3⋊D3—22no—
81C22×C3⋊D3C22⋊C411yescharacteristic
82C22×C3⋊D3C22⋊C411yescharacteristic
83C2×C32×C4C2×D411yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4#4112yes—
86C2×C32×C4#4112yes—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3C22⋊C411yescharacteristic
89C2×C3⋊Dic3C22⋊C411yescharacteristic
90C4×C3⋊D3—21no—
91C4×C3⋊D3—21no—
92C4×C3⋊D3—22no—
93C4×C3⋊D3—22no—
94C4×C3⋊D3—21no—
95C4×C3⋊D3—21no—
96C2×C3×Dic3—24no—
97C2×C3×Dic3—44no—
98#624—44no—
99#624—44no—
100C32×C8—41no—
101C3⋊(C3⋊C8)—41no—
102C3×C3⋊C8—42no—
103C22×C4⋊C4—91no—
104#407—91no—
105#407—91no—
106#343—32no—
107#343—92no—
108C23×D3—32no—
109C23×D3—61no—
110C22×C3×C4—21no—
111C22×C3×C4—34no—
112C22×C3×C4C22×D312yes—
113C22×Dic3—32no—
114C22×Dic3—61no—
115C22×Dic3C4×D314yes—
116C2×C4×D3—32no—
117C2×C4×D3—32no—
118C2×C4×D3—32no—
119C2×C4×D3—32no—
120C2×C4×D3—61no—
121C2×C4×D3—61no—
122C2×C4×D3—61no—
123C2×C4×D3—61no—
124C2×C4×D3—62no—
125C2×C4×D3—62no—
126C2×C4×D3—62no—
127C2×C4×D3—62no—
128C2×C4×D3—64no—
129C2×C4×D3—64no—
130C2×C4×D3—122no—
131C2×C4×D3—122no—
132C2×C4×D3—128no—
133C3×M16—22no—
134C3×M16—22no—
135C3×M16—42no—
136C3×M16—62no—
137C3×M16—62no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—62no—
141C3×C4⋊C4—62no—
142C3×C4⋊C4—62no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145Dic3⋊C4—64no—
146Dic3⋊C4—64no—
147#208—22no—
148#208—22no—
149#208—62no—
150#208—62no—
151#208—122no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—62no—
154C2×C3⋊C8—62no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32C2×C22⋊C411yescharacteristic
158C22×C32#11712yes—
159C2×C3⋊D3—28no—
160C2×C3⋊D3—21no—
161C2×C3⋊D3—21no—
162C2×C3⋊D3—21no—
163C2×C3⋊D3—21no—
164C2×C3⋊D3#10012yes—
165C2×C3⋊D3#10112yes—
166C32×C4#12311yescharacteristic
167C32×C4#12312yes—
168C32×C4#12311yescharacteristic
169C3⋊Dic3—21no—
170C3⋊Dic3—21no—
171C3×Dic3—44no—
172C23×C4—92no—
173C23×C4—91no—
174C2×M16—32no—
175C2×M16—91no—
176C2×M16D3211yescharacteristic
177C2×C4⋊C4—32no—
178C2×C4⋊C4—92no—
179C2×C4⋊C4—181no—
180C2×C4⋊C4—181no—
181C2×C4⋊C4—181no—
182C2×C4⋊C4—181no—
183C22⋊C8—181no—
184C22⋊C8—181no—
185C22⋊C8—181no—
186C22⋊C8—181no—
187C23×C3—21no—
188C23×C3C2×C4×D312yes—
189C22×D3—32no—
190C22×D3—32no—
191C22×D3—61no—
192C22×D3—61no—
193C22×D3—64no—
194C22×D3—64no—
195C22×D3—62no—
196C22×D3—62no—
197C22×D3—122no—
198C22×D3—122no—
199C22×D3—122no—
200C2×C3×C4—22no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—64no—
205C2×C3×C4—124no—
206C2×C3×C4#23512yes—
207C2×C3×C4#23512yes—
208C2×C3×C4D3×D412yes—
209C2×C3×C4D3×D412yes—
210C2×C3×C4#23712yes—
211C2×C3×C4#23912yes—
212C2×Dic3—24no—
213C2×Dic3—32no—
214C2×Dic3—32no—
215C2×Dic3—44no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—122no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—62no—
229C4×D3—122no—
230C4×D3—122no—
231C4×D3—121no—
232C4×D3—121no—
233C4×D3—121no—
234C4×D3—121no—
235C4×D3—128no—
236C4×D3—128no—
237C3×C8—42no—
238C3×C8—42no—
239C3×C8—122no—
240C3⋊C8—42no—
241C3⋊C8—122no—
242C3⋊C8—122no—
243C2×C32—22no—
244C2×C32#38111yescharacteristic
245C2×C32C2×#10011yescharacteristic
246C2×C32C2×#10111yescharacteristic
247C3⋊D3—22no—
248C3⋊D3—22no—
249C24—91no—
250C22×C4—34no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4—91no—
254C22×C4—91no—
255C22×C4—91no—
256C22×C4—182no—
257C22×C4—182no—
258C22×C4—182no—
259C22×C4—182no—
260C22×C4—364no—
261C22×C4C2×D3211yescharacteristic
262M16—22no—
263M16—62no—
264M16—62no—
265M16—182no—
266C2×C8—21no—
267C2×C8—62no—
268C2×C8—181no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—182no—
272C4⋊C4—182no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3#87212yes—
280C22×C3D3×C22⋊C412yes—
281C22×C3#89212yes—
282C2×D3—34no—
283C2×D3—34no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—68no—
287C2×D3—68no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—62no—
292C2×D3—121no—
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—121no—
296C2×D3—128no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—124no—
301C3×C4#88312yes—
302C3×C4#88312yes—
303C3×C4#89412yes—
304C3×C4#89412yes—
305Dic3—44no—
306Dic3—62no—
307Dic3—62no—
308Dic3—121no—
309Dic3—121no—
310C32#182111yescharacteristic, complemented, Sylow
311C23—91no—
312C23—91no—
313C23—182no—
314C23—182no—
315C23—182no—
316C23C4×D3211yescharacteristic
317C2×C4—64no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—124no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—364no—
329C2×C4—364no—
330C2×C4#379912yes—
331C2×C4#380211yescharacteristic
332C2×C4#380412yes—
333C2×C4#381111yescharacteristic
334C8—41no—
335C8—122no—
336C8—361no—
337C2×C3—22no—
338C2×C3—22no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—42no—
343C2×C3#466412yes—
344C2×C3D3×#10012yes—
345C2×C3D3×#10112yes—
346D3—64no—
347D3—64no—
348D3—122no—
349D3—122no—
350C22—21no—
351C22—21no—
352C22—92no—
353C22—92no—
354C22—188no—
355C22—181no—
356C22—181no—
357C22—181no—
358C22—181no—
359C22#6383612yes—
360C22#6392111yescharacteristic
361C4—124no—
362C4—181no—
363C4—181no—
364C4#6385412yes—
365C4#6389711yescharacteristic
366C4#6397511yescharacteristic
367C3—21no—
368C3#6899012yescomplemented
369C2—22no—
370C2—182no—
371C2—182no—
372C2#1059100111yescharacteristic
373C2#1059100311yescharacteristic
374C2#1059137111yescharacteristic
subgroupG/H
centerC22#63921
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C2×C22⋊C4
prime radicalC23×C3⋊D3C2×C4
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1821
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. C3×#4554C2semidirect
5. #10589167C2semidirect
28. #4554D3semidirect
310. C32#1821semidirect
368. C3#68990semidirect

Maximal subgroups

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

subgroupindex
C2×#639002
#105928252
#105910742
C3×#45542
#105891672
#689903