Finite Group Databasebeta

← #11922413

Subgroups — #11922413

Order 1152.

661 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 83 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×#1010⋊C3C211yesmaximal, characteristic
2#10593658C211yesmaximal, characteristic, complemented
3C3×#5426C211yesmaximal, characteristic, complemented
4D3×#859C212yesmaximal, complemented
5#10593209C212yesmaximal, complemented
6#85534—31nomaximal
7#78883—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#1010⋊C3C414yescomplemented
10C2×D3×SL(2,𝔽3)C2212yes—
11#63898—41nomaximal
12C3⋊#859C2211yescharacteristic
13C3⋊#859C2211yescharacteristic
14C3×#859C2212yes—
15C2×#1010D311yescharacteristic
16C2×2−1+4⋊C3—31no—
17C2×#907—31no—
18#5504—31no—
19#5426D311yescharacteristic, complemented
20#5424—31no—
21#5424—61no—
22#5420—32no—
23C2×#859—32no—
24C3×#417—31no—
25#5233—31no—
26#5120—31no—
27C2×#810—31no—
28#4802—32no—
29D3×#103—32no—
30#4772—32no—
31#4771—32no—
32C2×C3×D12—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×D3×Dic3—42no—
36D3×SL(2,𝔽3)C2×C414yescomplemented
37C4×C3⋊Dic3—41no—
38C3×C4⋊Dic3—41no—
39(C2×D3)⋊Dic3—42no—
40#2898—91noSylow
41C2×C3×#41C2×D311yescharacteristic
42#1010—34no—
43#1010Dic314yescomplemented
44C2×D3×Q8C2×D312yes—
45C2×#237—32no—
46C2×#235—31no—
47C22×Dic6—31no—
482−1+4⋊C3—34no—
49C2×#41⋊C3—21no—
50C2×#41⋊C3C2×D311yescharacteristic
51C22×SL(2,𝔽3)—32no—
52C2×C3×M16—31no—
53C2×C3×C4⋊C4—31no—
54D4×Dic3—121no—
55C4×#67—62no—
56#926—31no—
57C22×C3⋊C8—31no—
58#907—64no—
59C2×ΓL(1,𝔽25)—32no—
60C2×D3×C8—32no—
61#893—61no—
62#892—32no—
63D3×C4⋊C4—32no—
64#887—61no—
65#859—31no—
66#859—31no—
67#859—61no—
68#859—61no—
69#859C2×D312yes—
70C3×#103—32no—
71C3×#102—32no—
72#835—31no—
73#835—31no—
74#832—31no—
75#832—31no—
76#816—32no—
77#816—32no—
78#810—32no—
79#810—32no—
80C22×C3×D3—42no—
81C2×C32×C4—41no—
82C2×C3⋊Dic3—41no—
83C2×C3⋊Dic3—41no—
84C2×C3×Dic3—42no—
85C3×D12—44no—
86C3×SL(2,𝔽3)C22×C411yescharacteristic
87D3×Dic3—84no—
88C2×2−1+4—31no—
89C2×#131—91no—
90#520—91no—
91#419—92no—
92#419—91no—
93#417—31no—
94#417—92no—
95#416—92no—
96#416—91no—
97C2×#103—92no—
98C2×#103—91no—
99C3×#41—32no—
100C3×#41C2×Dic312yescomplemented
101C2×C3×Q8C22×D311yescharacteristic
102C2×C3×D4—31no—
103C2×C3×D4—121no—
104C22×C3×C4—31no—
105C22×C3×C4—61no—
106C2×#67—32no—
107C22×Dic3—61no—
108C22×Dic3—122no—
109D3×Q8—34no—
110D3×Q8C2×Dic314yescomplemented
111#237—38no—
112#237—64no—
113#235—34no—
114#235—34no—
115C2×D12S411yescharacteristic
116C2×C4×D3—32no—
117C2×C4×D3—62no—
118C2×Dic6—31no—
119C2×Dic6—34no—
120C2×Dic6—32no—
121C2×Dic6—62no—
122#41⋊C3—22no—
123#41⋊C3C4×D312yescomplemented
124C2×SL(2,𝔽3)—21no—
125C2×SL(2,𝔽3)—34no—
126C2×SL(2,𝔽3)C22×D311yescharacteristic
127C3×M16—62no—
128C2×C3×C8—32no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—61no—
131#217—61no—
132#217—122no—
133(C2×D3)⋊C4—62no—
134C4⋊Dic3—31no—
135C4⋊Dic3—31no—
136C4⋊Dic3—41no—
137Dic3⋊C4—62no—
138C4×Dic3—62no—
139C4×Dic3—61no—
140C4×Dic3—121no—
141C4×Dic3—241no—
142C2×C3⋊C8—31no—
143C2×C3⋊C8—31no—
144C2×C3⋊C8—62no—
145ΓL(1,𝔽25)—64no—
146D3×C8—64no—
147C22×C32—41no—
148C2×C3×D3—44no—
149C2×C3×D3—82no—
150C2×C3×D3—82no—
151C32×C4—42no—
152C3⋊Dic3—42no—
153C3⋊Dic3—81no—
154C3×Dic3—82no—
1552−1+4—34no—
1562−1+4—94no—
157C2×#41—91no—
158C2×#41—92no—
159C2×#41D3211yescharacteristic
160C22×Q8—32no—
161C22×Q8—91no—
162#131—184no—
163C2×M16—31no—
164C2×M16—92no—
165C22×C8—92no—
166C22×C8—91no—
167C4×Q8—181no—
168C4×D4—181no—
169C4×D4—182no—
170#117—92no—
171#117—91no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—92no—
174#103—32no—
175#103—92no—
176#103—92no—
177#103—92no—
178#103—92no—
179#103—91no—
180#103—91no—
181#102—32no—
182#102—91no—
183#102—91no—
184C23×C3—31no—
185C23×C3—122no—
186C22×D3C2×S412yes—
187C3×Q8—31no—
188C3×Q8C22×Dic311yescharacteristic
189C3×D4—32no—
190C3×D4—61no—
191C3×D4—124no—
192C2×C3×C4—31no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—41no—
196C2×C3×C4—62no—
197C2×C3×C4—81no—
198C2×C3×C4—241no—
199C2×C3×C4C2×S411yescharacteristic
200#67—38no—
201C2×Dic3—32no—
202C2×Dic3—42no—
203C2×Dic3—64no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—121no—
207C2×Dic3—121no—
208C2×Dic3—241no—
209C2×Dic3—241no—
210C2×Dic3—244no—
211D12SL(2,ℤ/4)14yes—
212C4×D3—34no—
213C4×D3—34no—
214C4×D3—242no—
215Dic6—34no—
216Dic6—34no—
217Dic6—62no—
218SL(2,𝔽3)—21no—
219SL(2,𝔽3)C2×C4×D311yescharacteristic
220C3×C8—62no—
221C3⋊C8—61no—
222C3⋊C8—61no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C3×D3—84no—
227#41—32no—
228#41—94no—
229#41—94no—
230#41—98no—
231#41—184no—
232#41D3×Dic312yescomplemented
233C2×Q8—34no—
234C2×Q8—92no—
235C2×Q8—94no—
236C2×Q8—91no—
237C2×Q8—94no—
238C2×Q8—182no—
239C2×Q8C2×D3211yescharacteristic
240C2×D4—31no—
241C2×D4—31no—
242C2×D4—92no—
243C22×C4—31no—
244C22×C4—61no—
245C22×C4—92no—
246C22×C4—182no—
247C22×C4—181no—
248M16—62no—
249M16—184no—
250C2×C8—32no—
251C2×C8—91no—
252C2×C8—91no—
253C2×C8—184no—
254C2×C8—182no—
255C4⋊C4—32no—
256C4⋊C4—61no—
257C4⋊C4—91no—
258C4⋊C4—91no—
259C4⋊C4—182no—
260C22⋊C4—182no—
261C22⋊C4—181no—
262C42—182no—
263C42—181no—
264C22×C3—32no—
265C22×C3—41no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—81no—
269C22×C3—124no—
270C22×C3—242no—
271C22×C3—242no—
272C22×C3C22×S411yescharacteristic
273C2×D3—22no—
274C2×D3—22no—
275C2×D3C2×SL(2,ℤ/4)14yes—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—42no—
279C3×C4—82no—
280C3×C4—122no—
281C3×C4C2×SL(2,ℤ/4)12yes—
282Dic3—34no—
283Dic3—82no—
284Dic3—122no—
285Dic3—121no—
286Dic3—121no—
287Dic3—241no—
288Dic3—242no—
289Dic3—242no—
290C32—41noSylow
291C23—31no—
292C23—32no—
293Q8—31no—
294Q8—94no—
295Q8—94no—
296Q8—182no—
297Q8C2×D3×Dic311yescharacteristic
298D4—32no—
299D4—34no—
300D4—61no—
301D4—98no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—62no—
306C2×C4—94no—
307C2×C4—94no—
308C2×C4—92no—
309C2×C4—184no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—241no—
313C2×C4—722no—
314C2×C4D3×S411yescharacteristic
315C8—62no—
316C8—181no—
317C8—181no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—62no—
322C2×C3—81no—
323C2×C3—81no—
324C2×C3—81no—
325C2×C3—244no—
326C2×C3#592611yescharacteristic
327C2×C3#592711yescharacteristic
328C2×C3C22×SL(2,ℤ/4)11yescharacteristic
329D3—24no—
330C22—32no—
331C22—34no—
332C22—62no—
333C22—61no—
334C22—61no—
335C22—62no—
336C22C2×D3×S411yescharacteristic
337C4—31no—
338C4—31no—
339C4—94no—
340C4—122no—
341C4—361no—
342C4—361no—
343C4D3×SL(2,ℤ/4)12yes—
344C3—41no—
345C3—81no—
346C3#8553411yescharacteristic, complemented
347C2—64no—
348C2—62no—
349C2#1059645311yescharacteristic
350C2#1059646011yescharacteristic
351C2C2×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×#1010⋊C3C2
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. #10593658C2semidirect
3. C3×#5426C2semidirect
4. D3×#859C2semidirect
5. #10593209C2semidirect
9. #1010⋊C3C4semidirect
19. #5426D3semidirect
33. C3×#41⋊C3C2×C4semidirect
36. D3×SL(2,𝔽3)C2×C4semidirect
43. #1010Dic3semidirect
100. C3×#41C2×Dic3semidirect
110. D3×Q8C2×Dic3semidirect
123. #41⋊C3C4×D3semidirect
232. #41D3×Dic3semidirect
346. C3#85534semidirect

Maximal subgroups

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

subgroupindex
C2×#1010⋊C32
#105936582
C3×#54262
D3×#8592
#105932092
#855343
#788833
#638984