Finite Group Databasebeta

← C2×#10593653 #11923050

Subgroups — C2×#10593653 #11923050

Order 1152.

884 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C22×C3⋊#226C211yesmaximal, characteristic, complemented
3#10593653C214yesmaximal, direct product factor
4C2×C3⋊#859C211yesmaximal, characteristic, complemented
5C2×#5419—33nomaximal
6C2×#5228—31nomaximal
7C22×C3×SL(2,𝔽3)—22no—
8C22×C3×SL(2,𝔽3)C2212yes—
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C2×C3⋊#226—22no—
11C2×C3⋊#226C2212yescomplemented
12C2×#3757—41nomaximal
13C3⋊#859C2212yescomplemented
14C23×C3×Q8D311yescharacteristic
15C23×SL(2,𝔽3)D313yes—
16C22×#226—33no—
17C22×#216—31no—
18C22×#216—31no—
19C2×#924—31no—
20#5419—312no—
21C2×#859—33no—
22#5228—34no—
23#5228—34no—
24C2×#835—31no—
25C2×#835—31no—
26C2×#830—31no—
27C24×C32—41no—
28C22×C3⋊Dic3—41no—
29C22×C3⋊Dic3—41no—
30C2×C3×SL(2,𝔽3)—22no—
31C2×C3×SL(2,𝔽3)D412yes—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊#226—22no—
34#3757—44no—
35C2×#451—91noSylow
36C22×C3×Q8—22no—
37C22×C3×Q8—31no—
38C22×C3×Q8—32no—
39C22×C3×Q8—62no—
40C22×C3×Q8C2×D312yes—
41C22×C3×Q8C2×D311yescharacteristic
42C23×C3×C4—31no—
43C22×Dic6—31no—
44C22×SL(2,𝔽3)—26no—
45C22×SL(2,𝔽3)C2×D316yes—
46C22×SL(2,𝔽3)C2×D313yes—
47C2×#226—36no—
48C2×#226—66no—
49C2×#217—61no—
50C2×#217—123no—
51C2×#216—32no—
52C2×#216—34no—
53C2×#216—32no—
54C2×#216—62no—
55C2×#216—64no—
56C2×#216—62no—
57C2×C4⋊Dic3—31no—
58#924—34no—
59#924—62no—
60C2×Dic3⋊C4—61no—
61C22×C3⋊C8—31no—
62C22×C3⋊C8—31no—
63#859—36no—
64#835—34no—
65#835—32no—
66#835—32no—
67#830—34no—
68C23×C32—44no—
69C23×C32—41no—
70C23×C32—42no—
71C23×C32—82no—
72C23×C32—82no—
73C2×C3⋊Dic3—42no—
74C2×C3⋊Dic3—42no—
75C2×C3⋊Dic3—82no—
76C2×C3⋊Dic3—82no—
77C3×SL(2,𝔽3)C2×D411yescharacteristic
78C23×Q8C3⋊D311yescharacteristic
79C22×Q16—91no—
80C22×Q16—91no—
81C2×C22⋊Q8—91no—
82#451—94no—
83#451—94no—
84C2×#103—91no—
85C2×#103—91no—
86C2×C22⋊C8—91no—
87C24×C3—43no—
88C24×C3S411yescharacteristic
89C2×C3×Q8—22no—
90C2×C3×Q8—31no—
91C2×C3×Q8—32no—
92C2×C3×Q8—32no—
93C2×C3×Q8—34no—
94C2×C3×Q8—61no—
95C2×C3×Q8—62no—
96C2×C3×Q8—64no—
97C2×C3×Q8—62no—
98C2×C3×Q8—64no—
99C2×C3×Q8#6712yes—
100C2×C3×Q8C22×D311yescharacteristic
101C22×C3×C4—31no—
102C22×C3×C4—32no—
103C22×C3×C4—31no—
104C22×C3×C4—32no—
105C22×C3×C4—62no—
106C22×C3×C4—62no—
107C22×Dic3—61no—
108C22×Dic3—61no—
109C22×Dic3—123no—
110C22×Dic3—123no—
111C2×Dic6—32no—
112C2×Dic6—62no—
113C2×Dic6—62no—
114C2×Dic6—61no—
115C2×SL(2,𝔽3)—26no—
116C2×SL(2,𝔽3)#6716yes—
117C2×SL(2,𝔽3)C22×D313yes—
118#226—66no—
119#217—64no—
120#217—1212no—
121#216—64no—
122#216—62no—
123#216—68no—
124#216—62no—
125C4⋊Dic3—32no—
126C4⋊Dic3—61no—
127Dic3⋊C4—64no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—32no—
130C2×C3⋊C8—62no—
131C2×C3⋊C8—62no—
132C22×C32—42no—
133C22×C32—44no—
134C22×C32—42no—
135C22×C32—42no—
136C22×C32—41no—
137C22×C32—82no—
138C22×C32—88no—
139C22×C32—82no—
140C3⋊Dic3—82no—
141C3⋊Dic3—82no—
142C22×Q8—22no—
143C22×Q8—31no—
144C22×Q8—32no—
145C22×Q8—62no—
146C22×Q8—91no—
147C22×Q8C2×C3⋊D312yes—
148C22×Q8C2×C3⋊D311yescharacteristic
149C23×C4—31no—
150C2×Q16—94no—
151C2×Q16—92no—
152C2×Q16—92no—
153C2×Q16—182no—
154C2×Q16—184no—
155C2×Q16—182no—
156C22×C8—91no—
157C22×C8—91no—
158C22⋊Q8—94no—
159C22⋊Q8—182no—
160C2×C4⋊C4—91no—
161C2×C4⋊C4—181no—
162C2×C22⋊C4—181no—
163#103—94no—
164#103—92no—
165#103—92no—
166C22⋊C8—94no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—46no—
170C23×C3—43no—
171C23×C3—412no—
172C23×C3—86no—
173C23×C3—86no—
174C23×C3#22614yescomplemented
175C23×C3C2×S412yes—
176C23×C3C2×S411yescharacteristic
177C3×Q8—32no—
178C3×Q8—31no—
179C3×Q8—34no—
180C3×Q8—61no—
181C3×Q8—62no—
182C3×Q8—64no—
183C3×Q8C2×#6711yescharacteristic
184C2×C3×C4—32no—
185C2×C3×C4—32no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—32no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—64no—
192C2×C3×C4—62no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—126no—
196C2×Dic3—126no—
197C2×Dic3—122no—
198C2×Dic3—122no—
199C2×Dic3—246no—
200C2×Dic3—246no—
201Dic6—62no—
202Dic6—62no—
203Dic6—122no—
204SL(2,𝔽3)C2×#6713yes—
205C3⋊C8—62no—
206C3⋊C8—62no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—44no—
211C2×C32—84no—
212C24C3⋊S411yescharacteristic
213C2×Q8—22no—
214C2×Q8—31no—
215C2×Q8—32no—
216C2×Q8—32no—
217C2×Q8—34no—
218C2×Q8—61no—
219C2×Q8—62no—
220C2×Q8—64no—
221C2×Q8—62no—
222C2×Q8—64no—
223C2×Q8—92no—
224C2×Q8—181no—
225C2×Q8—182no—
226C2×Q8—182no—
227C2×Q8#63812yes—
228C2×Q8C22×C3⋊D311yescharacteristic
229C22×C4—31no—
230C22×C4—32no—
231C22×C4—32no—
232C22×C4—31no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—181no—
236C22×C4—181no—
237Q16—184no—
238Q16—182no—
239Q16—188no—
240Q16—182no—
241C2×C8—92no—
242C2×C8—92no—
243C2×C8—182no—
244C2×C8—182no—
245C4⋊C4—92no—
246C4⋊C4—181no—
247C4⋊C4—184no—
248C22⋊C4—184no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—28no—
252C22×C3—46no—
253C22×C3—412no—
254C22×C3—46no—
255C22×C3—46no—
256C22×C3—43no—
257C22×C3—824no—
258C22×C3—86no—
259C22×C3—86no—
260C22×C3C2×#22612yes—
261C22×C3C2×#22614yescomplemented
262C22×C3#98312yes—
263C22×C3GL(2,ℤ/4)12yes—
264C22×C3C22×S411yescharacteristic
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—32no—
268C3×C4—62no—
269Dic3—122no—
270Dic3—122no—
271Dic3—246no—
272Dic3—246no—
273C32—41noSylow
274C23—22no—
275C23—22no—
276C23C3⋊#22614yescomplemented
277C23C2×C3⋊S412yes—
278C23C2×C3⋊S411yescharacteristic
279Q8—32no—
280Q8—31no—
281Q8—34no—
282Q8—61no—
283Q8—62no—
284Q8—64no—
285Q8—182no—
286Q8—182no—
287Q8—362no—
288Q8C2×#63811yescharacteristic
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—64no—
297C2×C4—62no—
298C2×C4—182no—
299C2×C4—182no—
300C2×C4—362no—
301C2×C4—362no—
302C8—182no—
303C8—182no—
304C2×C3—24no—
305C2×C3—412no—
306C2×C3—43no—
307C2×C3—43no—
308C2×C3—43no—
309C2×C3—812no—
310C2×C3#541914yescomplemented
311C2×C3C22×#22611yescharacteristic
312C2×C3C2×#98311yescharacteristic
313C2×C3C2×GL(2,ℤ/4)11yescharacteristic
314C22—28no—
315C22—22no—
316C22—22no—
317C22C2×C3⋊#22612yes—
318C22C2×C3⋊#22614yescomplemented
319C22#6426412yes—
320C22#6426912yes—
321C22C22×C3⋊S411yescharacteristic
322C4—31no—
323C4—31no—
324C4—32no—
325C4—62no—
326C4—362no—
327C4—362no—
328C3—43no—
329C3C2×#541911yescharacteristic, complemented
330C2—24no—
331C2#1059365314yesdirect product factor
332C2C22×C3⋊#22611yescharacteristic
333C2C2×#6426411yescharacteristic
334C2C2×#6426911yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC23×C3×Q8D3
socleC23×C3C2×S4
prime radicalC23×C3×SL(2,𝔽3)C2
hypercenterC24C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#226C2semidirect
3. #10593653C2direct (N × H)
4. C2×C3⋊#859C2semidirect
11. C2×C3⋊#226C22semidirect
13. C3⋊#859C22semidirect
174. C23×C3#226semidirect
261. C22×C3C2×#226semidirect
276. C23C3⋊#226semidirect
310. C2×C3#5419semidirect
318. C22C2×C3⋊#226semidirect
329. C3C2×#5419semidirect
331. C2#10593653direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C22×C3⋊#2262
#105936532
C2×C3⋊#8592
C2×#54193
C2×#52283
C2×#37574