Finite Group Databasebeta

← #11787246

Subgroups — #11787246

Order 1152.

726 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C4⋊#157C211yesmaximal, characteristic
2C3⋊D3×C22⋊Q8C211yesmaximal, characteristic
3C2×#63784C212yesmaximal, complemented
4#10593800C211yesmaximal, characteristic
5#10590163C212yesmaximal, complemented
6C23×#157—21no—
7C2×Q8×C3⋊D3C2211yescharacteristic
8C22×C4×C3⋊D3C2211yescharacteristic
9C2×C4⋊#157—21no—
10C2×C4⋊#157—24no—
11C2×C4⋊#157C2212yes—
12C2×#3787C2212yes—
13C32×C22⋊Q8C411yescharacteristic, complemented
14#64155C411yescharacteristic, complemented
15#64133C411yescharacteristic, complemented
16#64106C411yescharacteristic, complemented
17#64105—21no—
18C4⋊C4×C3⋊D3—21no—
19C4⋊C4×C3⋊D3C2211yescharacteristic
20C22⋊C4×C3⋊D3—21no—
21#64087—21no—
22C22⋊#622—21no—
23#63784—24no—
24#63778—21no—
25D3×C22⋊Q8—62no—
26C23×C3⋊D3D411yescharacteristic
27C22×#157—24no—
28C22×#157—22no—
29C22×#157—42no—
30C22×#157—42no—
31C2×C32×Q8C2×C411yescharacteristic, complemented
32C22×C32×C4C2×C411yescharacteristic
33C22×C3⋊Dic3D411yescharacteristic
34Q8×C3⋊D3—22no—
35Q8×C3⋊D3—22no—
36C2×C4×C3⋊D3—22no—
37C2×C4×C3⋊D3—22no—
38C2×C4×C3⋊D3—21no—
39C2×C4×C3⋊D3—21no—
40C2×C4×C3⋊D3C2×C411yescharacteristic
41C2×C4×C3⋊D3D411yescharacteristic
42C2×C4×C3⋊D3D411yescharacteristic
43C2×C4×C3⋊D3C2311yescharacteristic
44C2×#634—21no—
45C2×#634C2×C411yescharacteristic, complemented
46C2×#622—22no—
47C4⋊#157—24no—
48C4⋊#157—22no—
49C4⋊#157D414yes—
50#3787—22no—
51#3787—22no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2×C411yescharacteristic, complemented
54C32×C22⋊C4—21no—
55#3757—21no—
56#3752—21no—
57#3752—21no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C2×C411yescharacteristic, complemented
60#3750—21no—
61#3750—21no—
62#3750—21no—
63#1872—91nomaximal, Sylow
64C2×D3×Q8—62no—
65C22×C4×D3—62no—
66C3×C22⋊Q8—22no—
67#946—62no—
68#924—62no—
69#897—62no—
70#896—64no—
71D3×C4⋊C4—62no—
72D3×C4⋊C4—64no—
73D3×C22⋊C4—64no—
74#878—64no—
75C23×C32C22⋊C411yescharacteristic
76C22×C3⋊D3—21no—
77C22×C3⋊D3—21no—
78C22×C3⋊D3—21no—
79C22×C3⋊D3—21no—
80C22×C3⋊D3C22⋊C411yescharacteristic
81C22×C3⋊D3SD1612yes—
82C22×C3⋊D3Q1612yes—
83C22×C3⋊D3C2×D411yescharacteristic
84C2×#157—24no—
85C2×#157—48no—
86C2×#157—42no—
87C2×#157—42no—
88C32×Q8—22no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—22no—
92C2×C32×C4C22⋊C411yescharacteristic
93C2×C32×C4C22×C411yescharacteristic
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3C22⋊C411yescharacteristic
98C2×C3⋊Dic3C2×D411yescharacteristic
99C4×C3⋊D3—21no—
100C4×C3⋊D3—21no—
101C4×C3⋊D3—22no—
102C4×C3⋊D3—22no—
103C4×C3⋊D3—41no—
104C4×C3⋊D3—41no—
105C4×C3⋊D3—41no—
106C4×C3⋊D3—41no—
107C4×C3⋊D3C2×D412yes—
108C4×C3⋊D3#4112yes—
109#634—22no—
110#634—42no—
111#622—42no—
112C2×C22⋊Q8—91no—
113C22×C4⋊C4—91no—
114C2×#103—92no—
115C2×C22⋊C8—91no—
116#340—92no—
117C23×D3—62no—
118C2×C3×Q8—22no—
119C22×C3×C4—22no—
120C22×Dic3—62no—
121D3×Q8—124no—
122D3×Q8—124no—
123C2×C4×D3—62no—
124C2×C4×D3—62no—
125C2×C4×D3—62no—
126C2×C4×D3—62no—
127C2×C4×D3—64no—
128C2×C4×D3—64no—
129C2×C4×D3—124no—
130C2×C4×D3—124no—
131C2×Dic6—62no—
132C2×Dic6—64no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—22no—
135C3×C22⋊C4—24no—
136#217—64no—
137(C2×D3)⋊C4—64no—
138(C2×D3)⋊C4—64no—
139C4⋊Dic3—62no—
140C4⋊Dic3—64no—
141Dic3⋊C4—64no—
142Dic3⋊C4—64no—
143Dic3⋊C4—64no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32#10312yescomplemented
147C22×C32C2×C22⋊C411yescharacteristic
148C2×C3⋊D3—21no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3—22no—
151C2×C3⋊D3—22no—
152C2×C3⋊D3—22no—
153C2×C3⋊D3—22no—
154C2×C3⋊D3#10312yescomplemented
155C2×C3⋊D3#12011yescharacteristic
156C2×C3⋊D3#12311yescharacteristic
157C2×C3⋊D3C2×SD1611yescharacteristic
158C2×C3⋊D3C2×Q1611yescharacteristic
159C2×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
160C2×C3⋊D3#13711yescharacteristic
161#157—44no—
162C32×C4—21no—
163C32×C4—41no—
164C32×C4—41no—
165C32×C4C4×D412yes—
166C3⋊Dic3—21no—
167C3⋊Dic3—41no—
168C3⋊Dic3—41no—
169C3⋊Dic3#12112yes—
170C22×Q8—91no—
171C23×C4—91no—
172C23×C4—181no—
173C22×C8—92no—
174C22⋊Q8—91no—
175C22⋊Q8—91no—
176C22⋊Q8—91no—
177C22⋊Q8—181no—
178C22⋊Q8—181no—
179C22⋊Q8#15711yescharacteristic, complemented
180C2×C4⋊C4—92no—
181C2×C4⋊C4—91no—
182C2×C4⋊C4—181no—
183C2×C4⋊C4—184no—
184C2×C4⋊C4—181no—
185C2×C22⋊C4—181no—
186#103—184no—
187C22⋊C8—181no—
188C22⋊C8—181no—
189C23×C3—22no—
190C22×D3—62no—
191C22×D3—62no—
192C22×D3—64no—
193C22×D3—64no—
194C22×D3—122no—
195C22×D3—122no—
196C22×D3—122no—
197C22×D3—122no—
198C3×Q8—44no—
199C2×C3×C4—24no—
200C2×C3×C4—24no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—44no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—64no—
207C2×Dic3—64no—
208C2×Dic3—124no—
209C4×D3—64no—
210C4×D3—64no—
211C4×D3—124no—
212C4×D3—124no—
213C4×D3—122no—
214C4×D3—122no—
215C4×D3—124no—
216C4×D3—124no—
217C4×D3—124no—
218C4×D3—124no—
219Dic6—124no—
220Dic6—128no—
221C2×C32—22no—
222C2×C32#38611yescharacteristic
223C2×C32C2×#10311yescharacteristic
224C2×C32#41711yescharacteristic
225C3⋊D3—22no—
226C3⋊D3#45011yescharacteristic
227C3⋊D3#45111yescharacteristic
228C3⋊D3#48311yescharacteristic
229C3⋊D3#48411yescharacteristic
230C24—91no—
231C2×Q8—91no—
232C2×Q8—181no—
233C2×Q8—182no—
234C2×Q8—182no—
235C2×Q8C2×#15711yescharacteristic, complemented
236C22×C4—91no—
237C22×C4—91no—
238C22×C4—91no—
239C22×C4—91no—
240C22×C4—91no—
241C22×C4—182no—
242C22×C4—182no—
243C22×C4—181no—
244C22×C4—181no—
245C22×C4—182no—
246C22×C4—184no—
247C22×C4—362no—
248C22×C4—362no—
249C22×C4C2×#15711yescharacteristic
250C2×C8—182no—
251C2×C8—182no—
252C2×C8—182no—
253C4⋊C4—21no—
254C4⋊C4—94no—
255C4⋊C4—91no—
256C4⋊C4—181no—
257C4⋊C4—181no—
258C4⋊C4—181no—
259C4⋊C4—182no—
260C4⋊C4—184no—
261C4⋊C4—181no—
262C4⋊C4C2×#15711yescharacteristic, complemented
263C22⋊C4—21no—
264C22⋊C4—181no—
265C22⋊C4—181no—
266C22⋊C4—181no—
267C22×C3—22no—
268C22×C3—24no—
269C22×C3—42no—
270C22×C3—42no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—64no—
278C2×D3—124no—
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—124no—
282C2×D3—122no—
283C2×D3—122no—
284C3×C4—24no—
285C3×C4—42no—
286C3×C4—44no—
287C3×C4—44no—
288Dic3—64no—
289Dic3—122no—
290Dic3—124no—
291Dic3—124no—
292C32#187211yescharacteristic, complemented, Sylow
293C23—91no—
294C23—91no—
295C23—92no—
296C23—92no—
297C23—181no—
298C23—181no—
299C23—181no—
300C23—181no—
301C23C22⋊#15711yescharacteristic
302Q8—22no—
303Q8—182no—
304Q8—362no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—91no—
309C2×C4—92no—
310C2×C4—92no—
311C2×C4—91no—
312C2×C4—182no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—182no—
316C2×C4—182no—
317C2×C4—184no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—361no—
321C2×C4—361no—
322C2×C4—361no—
323C2×C4—361no—
324C2×C4—362no—
325C2×C4—362no—
326C2×C4—368no—
327C2×C4C22⋊#15711yescharacteristic
328C2×C4C22×#15711yescharacteristic
329C8—362no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—44no—
334D3—62no—
335D3—62no—
336D3—62no—
337D3—62no—
338D3—124no—
339C22—21no—
340C22—21no—
341C22—91no—
342C22—91no—
343C22—91no—
344C22—91no—
345C22—91no—
346C22—91no—
347C22—92no—
348C22—182no—
349C22—182no—
350C22—182no—
351C22—182no—
352C22—181no—
353C22—181no—
354C22#6378412yescomplemented
355C22C2×C22⋊#15711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—92no—
360C4—181no—
361C4—364no—
362C4—361no—
363C4—361no—
364C4D4×#15712yes—
365C3—22no—
366C2—22no—
367C2—91no—
368C2—91no—
369C2—91no—
370C2—91no—
371C2—182no—
372C2#1059382611yescharacteristic
373C2C2×#6378411yescharacteristic
374C2#1059384011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#157
commutatorC2×C32×C4C22×C4
FrattiniC2×C4C22×#157
FittingC32×C22⋊Q8C4
socleC22×C32C2×C22⋊C4
prime radicalC23×C3⋊D3D4
cosocleC2×C4×C3⋊D3C23
hypercenterC22⋊Q8#157
hypocenterC32#1872
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#63784C2semidirect
5. #10590163C2semidirect
13. C32×C22⋊Q8C4semidirect
14. #64155C4semidirect
15. #64133C4semidirect
16. #64106C4semidirect
31. C2×C32×Q8C2×C4semidirect
45. C2×#634C2×C4semidirect
53. C32×C4⋊C4C2×C4semidirect
59. C4⋊(C3⋊Dic3)C2×C4semidirect
146. C22×C32#103semidirect
154. C2×C3⋊D3#103semidirect
179. C22⋊Q8#157semidirect
235. C2×Q8C2×#157semidirect
262. C4⋊C4C2×#157semidirect
292. C32#1872semidirect
354. C22#63784semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#1572
C3⋊D3×C22⋊Q82
C2×#637842
#105938002
#105901632
#18729