Finite Group Databasebeta

← #11922387

Subgroups — #11922387

Order 1152.

589 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 67 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×#64266C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3C3×#5426C211yesmaximal, characteristic, complemented
4#10593214C212yesmaximal, complemented
5#10593195C212yesmaximal, complemented
6#85364—31nomaximal
7#78869—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64266C414yescomplemented
10C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊#226C2211yescharacteristic
12#63900—41nomaximal
13Dic3×SL(2,𝔽3)C2212yes—
14C3×#859C2212yes—
15C2×#985—31no—
16C2×#985—61no—
17#5827D311yescharacteristic, complemented
18C2×#950—31no—
19#5504—31no—
20#520⋊C3—31no—
21#5426D311yescharacteristic, complemented
22C4×GL(2,𝔽3)—32no—
23#5388—32no—
24C3×#417—31no—
25#5133—31no—
26#5122—31no—
27C2×#808—31no—
28#4789—32no—
29#4788—32no—
30#4785—32no—
31#4758—32no—
32C2×C4×C3⋊D3—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×#624—42no—
36C3⋊GL(2,𝔽3)C2×C412yescomplemented
37C3⋊#226C2×C412yescomplemented
38C3×C4⋊Dic3—41no—
39C3×C4⋊Dic3—41no—
40#3724—42no—
41#2952—91noSylow
42C2×C3×#41C2×D311yescharacteristic
43C2×#235—31no—
44C2×#41⋊C3—21no—
45C2×#41⋊C3C2×D311yescharacteristic
46#985—34no—
47#985—64no—
48C2×GL(2,𝔽3)—31no—
49C2×GL(2,𝔽3)—61no—
50C2×#226—31no—
51C2×#226—61no—
52C2×C3×M16—31no—
53C2×C3×C4⋊C4—31no—
54#950—34no—
55#950—34no—
56Q8×Dic3C2×D312yes—
57C2×#216—31no—
58C2×#215—31no—
59D4×Dic3—32no—
60C2×#214—31no—
61C2×#213—31no—
62C4×#67—62no—
63#926—31no—
64C2×C4⋊Dic3—31no—
65C22×C3⋊C8—31no—
66#893—32no—
67#892—61no—
68D3×C4⋊C4—61no—
69D3×C4⋊C4—121no—
70#887—32no—
71C4×SL(2,𝔽3)—32no—
72#859C2×D312yes—
73C3×#103—32no—
74C3×#102—32no—
75#821—32no—
76#816—32no—
77#815—32no—
78C8×Dic3—32no—
79#808—32no—
80#808—32no—
81C22×C3⋊D3—41no—
82C2×C32×C4—41no—
83C2×C3⋊Dic3—41no—
84C4×C3⋊D3—44no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—42no—
87C3×SL(2,𝔽3)C22×C411yescharacteristic
88#624—84no—
89C2×#135—91no—
90#520—31no—
91#520—91no—
92#442—92no—
93#441—92no—
94C4×SD16—92no—
95C4×SD16—92no—
96C2×#106—91no—
97#417—31no—
98#417—91no—
99#405—91no—
100C3×#41—32no—
101C3×#41C4×D312yescomplemented
102C2×C3×Q8C22×D311yescharacteristic
103C2×C3×D4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—61no—
106C2×#67—61no—
107C22×Dic3—61no—
108#235—34no—
109#235—62no—
110C2×D12—31no—
111C2×C4×D3—61no—
112C2×C4×D3—62no—
113C2×C4×D3—121no—
114C2×C4×D3—122no—
115C2×C4×D3—241no—
116C2×Dic6—31no—
117#41⋊C3—22no—
118#41⋊C3C4×D312yescomplemented
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)C22×D311yescharacteristic
121GL(2,𝔽3)—32no—
122GL(2,𝔽3)—62no—
123#226—32no—
124#226—62no—
125C3×M16—62no—
126C2×C3×C8—32no—
127C3×C4⋊C4—32no—
128C3×C4⋊C4—61no—
129C3×C4⋊C4—121no—
130#217—32no—
131#216—32no—
132#216—32no—
133#215—32no—
134#215—32no—
135#214—34no—
136#213—34no—
137(C2×D3)⋊C4—62no—
138C4⋊Dic3—31no—
139C4⋊Dic3—32no—
140C4⋊Dic3—41no—
141C4⋊Dic3S411yescharacteristic
142Dic3⋊C4—62no—
143Dic3⋊C4—122no—
144C4×Dic3—32no—
145C4×Dic3—62no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—34no—
149C22×C32—41no—
150C2×C3⋊D3—42no—
151C2×C3⋊D3—42no—
152C2×C3⋊D3—42no—
153C32×C4—42no—
154C3⋊Dic3—42no—
155C3×Dic3—82no—
156C3×Dic3—82no—
157C2×#41—91no—
158C2×#41D3211yescharacteristic
159#135—94no—
160#135—94no—
161C2×Q16—91no—
162C2×SD16—91no—
163C2×SD16—91no—
164C2×D8—91no—
165C2×M16—31no—
166C22×C8—91no—
167C4×Q8—32no—
168C4×Q8—92no—
169C4×D4—92no—
170C4×D4—92no—
171C4×D4—182no—
172#117—91no—
173#117—181no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—91no—
176C2×C4⋊C4—181no—
177#106—92no—
178#106—92no—
179#103—32no—
180#103—92no—
181#102—32no—
182#102—92no—
183#97—92no—
184C4×C8—92no—
185C23×C3—31no—
186C22×D3—61no—
187C22×D3—121no—
188C22×D3—241no—
189C3×Q8—31no—
190C3×Q8C2×C4×D311yescharacteristic
191C3×D4—32no—
192C3×D4—61no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—62no—
198C2×C3×C4—81no—
199C2×C3×C4—122no—
200C2×C3×C4—241no—
201C2×C3×C4C2×S411yescharacteristic
202#67—64no—
203C2×Dic3—32no—
204C2×Dic3—42no—
205C2×Dic3—61no—
206C2×Dic3—121no—
207C2×Dic3—241no—
208C2×Dic3—241no—
209C2×Dic3C2×S412yes—
210D12—32no—
211D12—61no—
212C4×D3—64no—
213C4×D3—124no—
214C4×D3—244no—
215C4×D3—242no—
216C4×D3—244no—
217Dic6—32no—
218Dic6—61no—
219SL(2,𝔽3)—21no—
220SL(2,𝔽3)C2×C4×D311yescharacteristic
221C3×C8—62no—
222C3⋊C8—32no—
223C3⋊C8—32no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C3⋊D3—44no—
228#41—32no—
229#41—94no—
230#41—182no—
231#41#62412yescomplemented
232C2×Q8—91no—
233C2×Q8C2×D3211yescharacteristic
234C2×D4—31no—
235C2×D4—91no—
236C2×D4—181no—
237C22×C4—31no—
238C22×C4—61no—
239C22×C4—182no—
240C22×C4—181no—
241C22×C4—181no—
242Q16—92no—
243Q16—92no—
244SD16—94no—
245SD16—92no—
246SD16—92no—
247D8—94no—
248M16—62no—
249C2×C8—32no—
250C2×C8—91no—
251C2×C8—91no—
252C2×C8—94no—
253C4⋊C4—31no—
254C4⋊C4—32no—
255C4⋊C4—61no—
256C4⋊C4—92no—
257C4⋊C4—91no—
258C4⋊C4—182no—
259C22⋊C4—92no—
260C22⋊C4—182no—
261C42—92no—
262C42—182no—
263C22×C3—32no—
264C22×C3—41no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—81no—
268C22×C3C22×S411yescharacteristic
269C2×D3—62no—
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—122no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—242no—
276C2×D3—242no—
277C2×D3—242no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—42no—
281C3×C4—82no—
282C3×C4—122no—
283C3×C4—242no—
284C3×C4C4×S412yes—
285Dic3—22no—
286Dic3—62no—
287Dic3—62no—
288Dic3—82no—
289Dic3—122no—
290Dic3—242no—
291C32—41noSylow
292C23—31no—
293C23—181no—
294Q8—31no—
295Q8—92no—
296Q8—181no—
297Q8C2×#62411yescharacteristic
298D4—32no—
299D4—61no—
300D4—92no—
301D4—181no—
302D4—184no—
303C2×C4—31no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—32no—
307C2×C4—62no—
308C2×C4—92no—
309C2×C4—181no—
310C2×C4—184no—
311C2×C4—241no—
312C2×C4—722no—
313C2×C4—721no—
314C2×C4D3×S411yescharacteristic
315C8—62no—
316C8—92no—
317C8—92no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—62no—
322C2×C3—81no—
323C2×C3—81no—
324C2×C3—81no—
325C2×C3C2×C4×S411yescharacteristic
326C2×C3#592411yescharacteristic
327C2×C3#592611yescharacteristic
328D3—124no—
329D3—122no—
330D3—244no—
331C22—32no—
332C22—61no—
333C22—61no—
334C22—182no—
335C22—361no—
336C22—361no—
337C22C2×D3×S411yescharacteristic
338C4—31no—
339C4—31no—
340C4—62no—
341C4—122no—
342C4—182no—
343C4—182no—
344C4(C3⋊S4)⋊C412yes—
345C3—41no—
346C3—81no—
347C3#8536411yescharacteristic, complemented
348C2—62no—
349C2—362no—
350C2#1059645011yescharacteristic
351C2#1059646011yescharacteristic
352C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×#64266C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC2×C4D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. C3×#5426C2semidirect
4. #10593214C2semidirect
5. #10593195C2semidirect
9. #64266C4semidirect
17. #5827D3semidirect
21. #5426D3semidirect
33. C3×#41⋊C3C2×C4semidirect
36. C3⋊GL(2,𝔽3)C2×C4semidirect
37. C3⋊#226C2×C4semidirect
101. C3×#41C4×D3semidirect
118. #41⋊C3C4×D3semidirect
231. #41#624semidirect
347. C3#85364semidirect

Maximal subgroups

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

subgroupindex
C2×#642662
#5827⋊C32
C3×#54262
#105932142
#105931952
#853643
#788693
#639004