Finite Group Databasebeta

← #11922365

Subgroups — #11922365

Order 1152.

617 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 71 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
2Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
3C3×#5424C211yesmaximal, characteristic, complemented
4#10593214C212yesmaximal, complemented
5#10593193C212yesmaximal, complemented
6C4×#985—31nomaximal
7#78818—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64266C414yescomplemented
10C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊#226C2211yescharacteristic
12C4×#624—41nomaximal
13Dic3×SL(2,𝔽3)C2212yes—
14C3×#859C2212yes—
15C2×#985—31no—
16C2×#985—61no—
17Dic3×#41D311yescharacteristic, complemented
18C2×#950—31no—
19#5522—31no—
20C4×#41⋊C3—31no—
21#5424D311yescharacteristic, complemented
22C4×GL(2,𝔽3)—32no—
23C4×#226—32no—
24C3×#416—31no—
25#5113—31no—
26C2×C8×Dic3—31no—
27#4998—31no—
28#4790—32no—
29#4788—32no—
30#4758—32no—
31#4756—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—
40Dic32—42no—
41C4×#135—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—
52C22×C3×C8—31no—
53C3×#117—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—
62#926—31no—
63C2×C4×Dic3—31no—
64C22×C3⋊C8—31no—
65#893—32no—
66#887—32no—
67#67⋊C4—62no—
68#872—61no—
69C42×D3—61no—
70C42×D3—121no—
71C4×SL(2,𝔽3)—32no—
72#859C2×D312yes—
73C3×#103—32no—
74C3×#102—32no—
75#821—32no—
76#816—32no—
77C8×Dic3—32no—
78C8×Dic3—32no—
79#808—32no—
80#807—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—48no—
89C2×#135—91no—
90C4×#41—31no—
91C4×#41—91no—
92C4×Q16—92no—
93C4×SD16—92no—
94C4×SD16—92no—
95C4×D8—92no—
96#427—91no—
97#416—31no—
98#416—91no—
99C2×C4×C8—91no—
100C3×#41—32no—
101C3×#41C4×D312yescomplemented
102C2×C3×Q8C22×D311yescharacteristic
103C2×C3×D4—31no—
104C22×C3×C4—31no—
105C2×#67—61no—
106C22×Dic3—32no—
107#235—34no—
108#235—62no—
109C2×D12—31no—
110C2×C4×D3—61no—
111C2×C4×D3—62no—
112C2×C4×D3—122no—
113C2×C4×D3—121no—
114C2×C4×D3—241no—
115C2×Dic6—31no—
116#41⋊C3—22no—
117#41⋊C3C4×D312yescomplemented
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)C22×D311yescharacteristic
120GL(2,𝔽3)—32no—
121GL(2,𝔽3)—62no—
122#226—32no—
123#226—62no—
124C2×C3×C8—32no—
125C2×C3×C8—62no—
126C3×C4⋊C4—32no—
127C3×C22⋊C4—61no—
128C3×C42—61no—
129C3×C42—121no—
130#217—61no—
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—32no—
139Dic3⋊C4—62no—
140C4×Dic3—31no—
141C4×Dic3—32no—
142C4×Dic3—41no—
143C4×Dic3—62no—
144C4×Dic3—122no—
145C4×Dic3S411yescharacteristic
146C2×C3⋊C8—31no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—34no—
149C22×C32—41no—
150C2×C3⋊D3—42no—
151C2×C3⋊D3—44no—
152C32×C4—42no—
153C3⋊Dic3—42no—
154C3×Dic3—44no—
155C3×Dic3—44no—
156C2×#41—91no—
157C2×#41D3211yescharacteristic
158#135—94no—
159#135—94no—
160C2×Q16—91no—
161C2×SD16—91no—
162C2×SD16—91no—
163C2×D8—91no—
164C22×C8—31no—
165C22×C8—91no—
166C4×Q8—32no—
167C4×Q8—92no—
168C4×D4—92no—
169C4×D4—92no—
170C4×D4—182no—
171#117—31no—
172#117—91no—
173#117—181no—
174C2×C42—91no—
175C2×C42—181no—
176#107—92no—
177#106—92no—
178#103—32no—
179#103—92no—
180#102—32no—
181#102—92no—
182C4×C8—92no—
183C4×C8—92no—
184C23×C3—31no—
185C22×D3—61no—
186C22×D3—121no—
187C22×D3—241no—
188C3×Q8—31no—
189C3×Q8C2×C4×D311yescharacteristic
190C3×D4—32no—
191C3×D4—61no—
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—122no—
199C2×C3×C4C2×S411yescharacteristic
200#67—64no—
201C2×Dic3—32no—
202C2×Dic3—42no—
203C2×Dic3—61no—
204C2×Dic3—64no—
205C2×Dic3—121no—
206C2×Dic3—241no—
207C2×Dic3C2×S412yes—
208D12—32no—
209D12—61no—
210C4×D3—64no—
211C4×D3—124no—
212C4×D3—128no—
213C4×D3—124no—
214C4×D3—244no—
215Dic6—32no—
216Dic6—61no—
217SL(2,𝔽3)—21no—
218SL(2,𝔽3)C2×C4×D311yescharacteristic
219C3×C8—62no—
220C3⋊C8—32no—
221C3⋊C8—32no—
222C2×C32—41no—
223C2×C32—42no—
224C3⋊D3—44no—
225#41—32no—
226#41—94no—
227#41—182no—
228#41#62412yescomplemented
229C2×Q8—91no—
230C2×Q8C2×D3211yescharacteristic
231C2×D4—31no—
232C2×D4—91no—
233C2×D4—181no—
234C22×C4—31no—
235C22×C4—92no—
236C22×C4—182no—
237C22×C4—181no—
238Q16—92no—
239Q16—92no—
240SD16—94no—
241SD16—92no—
242SD16—92no—
243D8—94no—
244C2×C8—32no—
245C2×C8—62no—
246C2×C8—91no—
247C2×C8—91no—
248C2×C8—94no—
249C4⋊C4—32no—
250C4⋊C4—92no—
251C4⋊C4—182no—
252C22⋊C4—61no—
253C22⋊C4—182no—
254C22⋊C4—181no—
255C42—31no—
256C42—61no—
257C42—92no—
258C42—91no—
259C42—182no—
260C22×C3—32no—
261C22×C3—41no—
262C22×C3—62no—
263C22×C3—81no—
264C22×C3C22×S411yescharacteristic
265C2×D3—62no—
266C2×D3—122no—
267C2×D3—124no—
268C2×D3—122no—
269C2×D3—242no—
270C2×D3—244no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—42no—
274C3×C4—82no—
275C3×C4—124no—
276C3×C4—122no—
277C3×C4C4×S412yes—
278Dic3—44no—
279Dic3—62no—
280Dic3—62no—
281Dic3—122no—
282Dic3—242no—
283Dic3#98514yescomplemented
284C32—41noSylow
285C23—31no—
286C23—181no—
287Q8—31no—
288Q8—92no—
289Q8—181no—
290Q8C2×#62411yescharacteristic
291D4—32no—
292D4—61no—
293D4—92no—
294D4—181no—
295D4—184no—
296C2×C4—32no—
297C2×C4—31no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—62no—
301C2×C4—92no—
302C2×C4—184no—
303C2×C4—181no—
304C2×C4—184no—
305C2×C4—364no—
306C2×C4D3×S411yescharacteristic
307C8—62no—
308C8—92no—
309C8—92no—
310C2×C3—42no—
311C2×C3—41no—
312C2×C3—62no—
313C2×C3—82no—
314C2×C3—81no—
315C2×C3C2×C4×S411yescharacteristic
316C2×C3C2×#98512yes—
317D3—124no—
318D3—122no—
319D3—244no—
320C22—32no—
321C22—62no—
322C22—182no—
323C22—362no—
324C22C2×D3×S411yescharacteristic
325C4—34no—
326C4—31no—
327C4—31no—
328C4—122no—
329C4—182no—
330C4—182no—
331C4(C3⋊S4)⋊C412yes—
332C3—41no—
333C3—81no—
334C3C4×#98511yescharacteristic, complemented
335C2—62no—
336C2—362no—
337C2D3×#98512yes—
338C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C2×D3×S4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×#64266C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. Dic3×#41⋊C3C2semidirect
3. C3×#5424C2semidirect
4. #10593214C2semidirect
5. #10593193C2semidirect
9. #64266C4semidirect
17. Dic3×#41D3semidirect
21. #5424D3semidirect
33. C3×#41⋊C3C2×C4semidirect
36. C3⋊GL(2,𝔽3)C2×C4semidirect
37. C3⋊#226C2×C4semidirect
101. C3×#41C4×D3semidirect
117. #41⋊C3C4×D3semidirect
228. #41#624semidirect
283. Dic3#985semidirect
334. C3C4×#985semidirect

Maximal subgroups

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

subgroupindex
C2×#642662
Dic3×#41⋊C32
C3×#54242
#105932142
#105931932
C4×#9853
#788183
C4×#6244