Finite Group Databasebeta

← #11798727

Subgroups — #11798727

Order 1152.

528 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C3×#5672C211yesmaximal, characteristic, complemented
2#10592760C211yesmaximal, characteristic
3#10591535C212yesmaximal
4C3×#4541C211yesmaximal, characteristic, complemented
5#10589175C212yesmaximal, complemented
6#71958—31nomaximal
7#69155—31nomaximal
8#64361C2211yescharacteristic
9C32×#124C2211yescharacteristic
10#64151—21no—
11C3×#944—21no—
12C3×#935C2212yes—
13C3×#879—22no—
14C3×#878—22no—
15C3×#870—21no—
16#63966—22no—
17#63660C2211yescharacteristic
18#63632—21no—
19C3×#834C2212yes—
20#63581—22no—
21C3×#563—31no—
22#5672D311yescharacteristic, complemented
23#5061—31no—
24#5061—31no—
25#5061—61no—
26#5053—32no—
27#4742—32no—
28C3×#354—31no—
29#4541D311yescharacteristic, complemented
30#4539—32no—
31#4472—32no—
32C2×C32×Q8D411yescharacteristic
33C2×C32×D4C2311yescharacteristic
34C2×#638—22no—
35#3830—41no—
36D4×C3⋊D3—41no—
37#3828—21no—
38C22×C3×Dic3D412yes—
39C2×C3×Dic6—21no—
40C32×C22⋊C4—22no—
41C32×C42D411yescharacteristic
42#3756—41no—
43#3755—41no—
44#3748—21no—
45C3×#217—22no—
46C3×#217—22no—
47C3×#217D412yes—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×Dic3⋊C4—22no—
51C3×Dic3⋊C4—22no—
52C3×Dic3⋊C4—22no—
53C3×C4×Dic3—21no—
54Dic3⋊Dic3—42no—
55#3722—42no—
56#2201—91noSylow
57#1004—31no—
58#1004—31no—
59#1004—61no—
60C3×C4⋊Q8—61no—
61C3×#126—62no—
62C3×#125—61no—
63C3×#124—21no—
64C3×#124C2×D311yescharacteristic
65C3×#124C2×D311yescharacteristic
66C3×#123—32no—
67C3×C22⋊Q8—62no—
68#944—21no—
69#942—62no—
70#942—61no—
71#942—61no—
72#935C2×D312yes—
73#929—62no—
74#886—62no—
75#879—22no—
76#878—22no—
77#870—21no—
78C3×#99—32no—
79#834C2×D312yes—
80#833—31no—
81#833—31no—
82#833—61no—
83#806—62no—
84#806—62no—
85#805—62no—
86#805—61no—
87#805—61no—
88C23×C32C2×D412yes—
89C22×C3⋊D3—22no—
90C32×Q8—41no—
91C32×D4—41no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4C2×D411yescharacteristic
95#638—42no—
96#638—41no—
97C2×C3⋊Dic3—22no—
98C4⋊(C3⋊D3)—21no—
99C4×C3⋊D3—41no—
100#634—21no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—42no—
104C2×C3×Dic3—42no—
105C3×Dic6—81no—
106C3⋊(C3⋊C8)—41no—
107#563—31no—
108#458—92no—
109#455—91no—
110#354—31no—
111#352—92no—
112C2×C3×Q8—21no—
113C2×C3×Q8—61no—
114C2×C3×Q8D1211yescharacteristic
115C2×C3×Q8#6711yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4C22×D311yescharacteristic
118C2×C3×D4C22×D311yescharacteristic
119C22×C3×C4—32no—
120C2×#67—62no—
121C2×#67—62no—
122C2×#67—122no—
123C22×Dic3#6712yes—
124#237—121no—
125#237—121no—
126#237—122no—
127D3×D4—121no—
128D3×D4—121no—
129D3×D4—122no—
130#235—62no—
131#235—61no—
132#235—61no—
133C2×Dic6—21no—
134C3×C4⋊C4—62no—
135C3×C4⋊C4—62no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—62no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—32no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—62no—
144C3×C22⋊C4—62no—
145C3×C42—21no—
146C3×C42—61no—
147C3×C42D1211yescharacteristic
148C3×C42#6711yescharacteristic
149#217—22no—
150#217—22no—
151#217#6712yes—
152#216—121no—
153#216—121no—
154#216—122no—
155#215—121no—
156#215—121no—
157#215—122no—
158(C2×D3)⋊C4—122no—
159(C2×D3)⋊C4—122no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—122no—
163Dic3⋊C4—22no—
164Dic3⋊C4—22no—
165Dic3⋊C4—22no—
166Dic3⋊C4—122no—
167C4×Dic3—21no—
168#208—62no—
169#208—61no—
170#208—61no—
171C22×C32—22no—
172C22×C32—42no—
173C22×C32#12011yescharacteristic
174C2×C3⋊D3—41no—
175C2×C3⋊D3—82no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—41no—
179C3⋊Dic3—41no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—82no—
183H(5,𝔽2)—91no—
184#137—181no—
185C4⋊Q8—61no—
186#126—62no—
187#125—61no—
188#124D3211yescharacteristic
189#123—32no—
190#123—182no—
191C22⋊Q8—62no—
192#104—181no—
193#100—91no—
194#99—32no—
195#99—182no—
196C23×C3—22no—
197C23×C3D3×D412yes—
198C23×C3C2×#6712yes—
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—122no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—42no—
205C3×Q8—241no—
206C3×D4—41no—
207C3×D4—41no—
208C3×D4—42no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—122no—
219C2×C3×C4—122no—
220C2×C3×C4C2×D1211yescharacteristic
221C2×C3×C4C2×#6711yescharacteristic
222#67—122no—
223#67—122no—
224#67—122no—
225#67—124no—
226#67—121no—
227#67—121no—
228C2×Dic3—22no—
229C2×Dic3—22no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—122no—
235D12—62no—
236D12—61no—
237D12—61no—
238C4×D3—122no—
239C4×D3—121no—
240C4×D3—121no—
241Dic6—62no—
242Dic6—61no—
243Dic6—61no—
244Dic6—81no—
245C3⋊C8—122no—
246C3⋊C8—121no—
247C3⋊C8—121no—
248C2×C32—21no—
249C2×C32—42no—
250C2×C32B(4,𝔽2)11yescharacteristic
251C3⋊D3—81no—
252#41—181no—
253#41—361no—
254C2×Q8—61no—
255C2×Q8#62611yescharacteristic
256C2×D4—182no—
257C2×D4—361no—
258C2×D4C2×D3211yescharacteristic
259C22×C4—32no—
260Q16—361no—
261SD16—361no—
262M16—181no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—362no—
269C22⋊C4—22no—
270C22⋊C4—32no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C22⋊C4—362no—
274C42—61no—
275C42#62611yescharacteristic
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—21no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—42no—
282C22×C3—82no—
283C22×C3#88211yescharacteristic
284C22×C3#95311yescharacteristic
285C2×D3—122no—
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—242no—
289C2×D3—242no—
290C2×D3—482no—
291C3×C4—22no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—122no—
301C3×C4—122no—
302C3×C4—242no—
303Dic3—42no—
304Dic3—42no—
305Dic3—82no—
306Dic3—122no—
307Dic3—121no—
308Dic3—121no—
309C32#220111yescharacteristic, complemented, Sylow
310C23—182no—
311C23D3×#6712yes—
312Q8—41no—
313Q8—181no—
314Q8—241no—
315D4—41no—
316D4—181no—
317D4—362no—
318D4—361no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—182no—
326C2×C4—361no—
327C2×C4C2×#62611yescharacteristic
328C8—361no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—41no—
333C2×C3—42no—
334C2×C3—42no—
335C2×C3—82no—
336C2×C3#474111yescharacteristic
337C2×C3#524411yescharacteristic
338D3—242no—
339D3—241no—
340D3—241no—
341C22—22no—
342C22—42no—
343C22—361no—
344C22—722no—
345C22#6397711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—122no—
350C4—122no—
351C4—242no—
352C4—361no—
353C3—21no—
354C3#6915511yescharacteristic, complemented
355C3#7195811yescharacteristic, complemented
356C2—21no—
357C2—42no—
358C2—721no—
359C2#1059153111yescharacteristic
subgroupG/H
centerC2#10591531
commutator, cosocleC2×C32×D4C23
FrattiniC2×D4C2×D32
FittingC32×#124C22
socleC2×C32B(4,𝔽2)
prime radical#64361C22
hypercenter#124D32
hypocenterC32#2201
LCSG ▷ C2×C32×D4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×#5672C2semidirect
4. C3×#4541C2semidirect
5. #10589175C2semidirect
22. #5672D3semidirect
29. #4541D3semidirect
309. C32#2201semidirect
354. C3#69155semidirect
355. C3#71958semidirect

Maximal subgroups

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

subgroupindex
C3×#56722
#105927602
#105915352
C3×#45412
#105891752
#719583
#691553