Finite Group Databasebeta

← C2×#10593208 #11922114

Subgroups — C2×#10593208 #11922114

Order 1152.

693 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 75 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×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C22×C3×#226C211yesmaximal, characteristic, complemented
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593208C214yesmaximal, direct product factor
5C2×#5395—31nomaximal
6C2×#5185—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊GL(2,𝔽3)—22no—
9C2×C3⋊GL(2,𝔽3)C2212yes—
10C2×C3×#226—22no—
11C2×C3×#226C2212yescomplemented
12C2×#3722—41nomaximal
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C22×GL(2,𝔽3)—31no—
15C22×GL(2,𝔽3)—61no—
16C22×#226D311yescharacteristic, complemented
17C22×C3×Q16—31no—
18C2×#947—31no—
19C2×Q8×Dic3D311yescharacteristic, complemented
20C22×#215—31no—
21C2×C4×SL(2,𝔽3)—31no—
22#5395—34no—
23C2×#835—31no—
24#5185—34no—
25#5185—34no—
26C2×#821—31no—
27C2×Dic3⋊C8—31no—
28C23×C3⋊D3—41no—
29C22×C3×Dic3—41no—
30C22×C3×Dic3—41no—
31C2×C3×SL(2,𝔽3)D412yes—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)—22no—
34C3×#226—22no—
35#3722—44no—
36C2×#464—91noSylow
37C22×C3×Q8—31no—
38C22×C3×Q8C2×D311yescharacteristic
39C22×D12—31no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)C2×D311yescharacteristic
42C2×GL(2,𝔽3)—32no—
43C2×GL(2,𝔽3)—62no—
44C2×GL(2,𝔽3)—64no—
45C2×GL(2,𝔽3)—62no—
46C2×#226—22no—
47C2×#226C2×D312yescomplemented
48C2×C3×Q16—32no—
49C2×C3×Q16—32no—
50C2×C3×Q16—62no—
51C2×C3×Q16—62no—
52C22×C3×C8—31no—
53#947—34no—
54#947—62no—
55Q8×Dic3—32no—
56Q8×Dic3C2×D312yescomplemented
57C2×#215—32no—
58C2×#215—32no—
59C2×#215—62no—
60C2×#215—62no—
61C2×(C2×D3)⋊C4—61no—
62C2×(C2×D3)⋊C4—61no—
63C2×(C2×D3)⋊C4—121no—
64C2×C4⋊Dic3—31no—
65C2×C4×Dic3—31no—
66C22×C3⋊C8—31no—
67C4×SL(2,𝔽3)—32no—
68#835—34no—
69#821—34no—
70Dic3⋊C8—34no—
71C23×C32—41no—
72C22×C3⋊D3—42no—
73C22×C3⋊D3—44no—
74C22×C3⋊D3—82no—
75C22×C3⋊D3—82no—
76C2×C3×Dic3—42no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—82no—
79C2×C3×Dic3—82no—
80C3×SL(2,𝔽3)C2×D411yescharacteristic
81C22×Q16—31no—
82C22×SD16—91no—
83C2×#124—91no—
84C2×C4×Q8—31no—
85#464—94no—
86#464—94no—
87C2×C4⋊C8—91no—
88C2×#103—91no—
89C2×#102—91no—
90C23×D3—61no—
91C23×D3—121no—
92C23×D3—241no—
93C2×C3×Q8—31no—
94C2×C3×Q8—32no—
95C2×C3×Q8—32no—
96C2×C3×Q8—61no—
97C2×C3×Q8—62no—
98C2×C3×Q8—62no—
99C2×C3×Q8D1212yes—
100C2×C3×Q8C22×D311yescharacteristic
101C22×C3×C4—31no—
102C22×C3×C4—61no—
103C22×C3×C4—121no—
104C22×Dic3—31no—
105C22×Dic3—41no—
106C22×Dic3S411yescharacteristic
107C2×D12—32no—
108C2×D12—62no—
109C2×D12—62no—
110C2×D12—61no—
111C2×SL(2,𝔽3)—22no—
112C2×SL(2,𝔽3)—21no—
113C2×SL(2,𝔽3)#6712yes—
114C2×SL(2,𝔽3)C22×D311yescharacteristic
115GL(2,𝔽3)—64no—
116GL(2,𝔽3)—62no—
117#226—22no—
118C3×Q16—64no—
119C3×Q16—62no—
120C3×Q16—62no—
121C2×C3×C8—32no—
122C2×C3×C8—62no—
123#215—64no—
124#215—62no—
125#215—62no—
126(C2×D3)⋊C4—64no—
127(C2×D3)⋊C4—64no—
128(C2×D3)⋊C4—124no—
129C4⋊Dic3—32no—
130C4⋊Dic3—61no—
131C4×Dic3—32no—
132C4×Dic3—32no—
133C2×C3⋊C8—32no—
134C2×C3⋊C8—62no—
135C22×C32—42no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—41no—
139C2×C3⋊D3—44no—
140C2×C3⋊D3—82no—
141C2×C3⋊D3—82no—
142C2×C3⋊D3—88no—
143C3×Dic3—82no—
144C3×Dic3—82no—
145C22×Q8—31no—
146C22×Q8D3211yescharacteristic
147C22×D4—91no—
148C2×Q16—32no—
149C2×Q16—32no—
150C2×Q16—62no—
151C2×Q16—62no—
152C2×SD16—92no—
153C2×SD16—92no—
154C2×SD16—182no—
155C2×SD16—182no—
156C22×C8—31no—
157C22×C8—91no—
158#124—94no—
159#124—182no—
160C4×Q8—32no—
161C4×Q8—92no—
162C2×C4⋊C4—91no—
163C2×C22⋊C4—181no—
164C2×C22⋊C4—181no—
165C2×C42—91no—
166C4⋊C8—94no—
167#103—94no—
168#102—94no—
169C23×C3—41no—
170C23×C3—81no—
171C23×C3C2×S411yescharacteristic
172C22×D3—62no—
173C22×D3—122no—
174C22×D3—124no—
175C22×D3—122no—
176C22×D3—122no—
177C22×D3—122no—
178C22×D3—242no—
179C22×D3—242no—
180C22×D3—242no—
181C22×D3—244no—
182C22×D3—244no—
183C22×D3—244no—
184C3×Q8—32no—
185C3×Q8—31no—
186C3×Q8—61no—
187C3×Q8—62no—
188C3×Q8—62no—
189C3×Q8—122no—
190C3×Q8C2×D1211yescharacteristic
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—62no—
196C2×C3×C4—122no—
197C2×C3×C4—122no—
198C2×C3×C4—242no—
199C2×Dic3—22no—
200C2×Dic3—32no—
201C2×Dic3—42no—
202C2×Dic3—62no—
203C2×Dic3—82no—
204C2×Dic3C2×S412yes—
205D12—62no—
206D12—62no—
207D12—122no—
208SL(2,𝔽3)—21no—
209SL(2,𝔽3)C2×#6711yescharacteristic
210C3×C8—62no—
211C3⋊C8—62no—
212C2×C32—44no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C3⋊D3—84no—
217C24—181no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—32no—
221C2×Q8—61no—
222C2×Q8—62no—
223C2×Q8—62no—
224C2×Q8#62612yes—
225C2×Q8C2×D3211yescharacteristic
226C2×D4—92no—
227C2×D4—182no—
228C2×D4—182no—
229C2×D4—181no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—91no—
234Q16—64no—
235Q16—62no—
236Q16—62no—
237SD16—184no—
238SD16—182no—
239SD16—182no—
240C2×C8—32no—
241C2×C8—62no—
242C2×C8—92no—
243C2×C8—182no—
244C4⋊C4—92no—
245C4⋊C4—181no—
246C22⋊C4—184no—
247C22⋊C4—184no—
248C42—92no—
249C42—92no—
250C22×C3—41no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—81no—
255C22×C3—82no—
256C22×C3—82no—
257C22×C3—82no—
258C22×C3C4⋊S412yes—
259C22×C3#98312yes—
260C22×C3#98512yes—
261C22×C3C22×S411yescharacteristic
262C2×D3—124no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—122no—
266C2×D3—242no—
267C2×D3—242no—
268C2×D3—244no—
269C2×D3—244no—
270C2×D3—244no—
271C2×D3—248no—
272C2×D3—2416no—
273C2×D3—244no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—32no—
277C3×C4—122no—
278C3×C4—242no—
279Dic3—22no—
280Dic3—62no—
281Dic3—82no—
282C32—41noSylow
283C23—182no—
284C23—362no—
285C23—362no—
286C23—362no—
287C23D3×S411yescharacteristic
288Q8—32no—
289Q8—31no—
290Q8—61no—
291Q8—62no—
292Q8—62no—
293Q8—122no—
294Q8C2×#62611yescharacteristic
295D4—182no—
296D4—182no—
297D4—362no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—92no—
306C2×C4—122no—
307C2×C4—182no—
308C8—62no—
309C8—182no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—44no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3—81no—
317C2×C3—84no—
318C2×C3#539514yescomplemented
319C2×C3C2×C4⋊S411yescharacteristic
320C2×C3C2×#98311yescharacteristic
321C2×C3C2×#98511yescharacteristic
322D3—244no—
323D3—248no—
324D3—242no—
325C22—362no—
326C22—362no—
327C22—362no—
328C22—724no—
329C22#6419812yes—
330C22#6420212yes—
331C22#6420712yes—
332C22C2×D3×S411yescharacteristic
333C4—31no—
334C4—31no—
335C4—32no—
336C4—62no—
337C4—122no—
338C4—182no—
339C3—41no—
340C3—81no—
341C3C2×#539511yescharacteristic, complemented
342C2—722no—
343C2#1059320814yesdirect product factor
344C2C2×#6419811yescharacteristic
345C2C2×#6420211yescharacteristic
346C2C2×#6420711yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C2×D3×S4
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radicalC22×C3⋊GL(2,𝔽3)C2
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×#226C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. #10593208C2direct (N × H)
11. C2×C3×#226C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
16. C22×#226D3semidirect
19. C2×Q8×Dic3D3semidirect
47. C2×#226C2×D3semidirect
56. Q8×Dic3C2×D3semidirect
318. C2×C3#5395semidirect
341. C3C2×#5395semidirect
343. C2#10593208direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊GL(2,𝔽3)2
C22×C3×#2262
C2×Dic3×SL(2,𝔽3)2
#105932082
C2×#53953
C2×#51853
C2×#37224