Finite Group Databasebeta

← C2×#10593204 #11922109

Subgroups — C2×#10593204 #11922109

Order 1152.

689 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 91 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
2C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593204C214yesmaximal, direct product factor
5C2×#5394—31nomaximal
6C2×#5164—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×GL(2,𝔽3)C414yescomplemented
9C2×C3×GL(2,𝔽3)C2212yes—
10C2×(C2×D3)⋊Dic3—41nomaximal
11Dic3×SL(2,𝔽3)C2212yescomplemented
12C3⋊#859C2212yescomplemented
13C22×GL(2,𝔽3)D311yescharacteristic
14C22×C3×SD16—31no—
15C2×Q8×Dic3D311yescharacteristic, complemented
16C2×D4×Dic3—31no—
17C2×C4×SL(2,𝔽3)—31no—
18C2×#859—31no—
19C2×#859—61no—
20#5394—34no—
21C2×#835—31no—
22C2×#832—31no—
23#5164—34no—
24#5164—34no—
25C2×#818—31no—
26C2×#815—31no—
27C23×C3×D3—41no—
28C22×C3⋊Dic3—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3×GL(2,𝔽3)C2×C414yescomplemented
33(C2×D3)⋊Dic3—44no—
34C2×#440—91noSylow
35C22×C3×Q8C2×D311yescharacteristic
36C22×C3×D4—31no—
37C23×Dic3—61no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)C2×D311yescharacteristic
40C2×GL(2,𝔽3)Dic314yescomplemented
41C2×GL(2,𝔽3)C2×D312yes—
42C2×C3×SD16—34no—
43C2×C3×SD16—32no—
44C2×C3×SD16—34no—
45C2×C3×SD16—32no—
46C22×C3×C8—31no—
47C2×#217—61no—
48Q8×Dic3—32no—
49Q8×Dic3C2×D312yescomplemented
50D4×Dic3—34no—
51D4×Dic3—62no—
52C2×(C2×D3)⋊C4—121no—
53C2×C4⋊Dic3—31no—
54C2×C4⋊Dic3—31no—
55C2×C4×Dic3—31no—
56C22×C3⋊C8—31no—
57C4×SL(2,𝔽3)—32no—
58#859—32no—
59#859—62no—
60#835—32no—
61#835—32no—
62#832—34no—
63#818—34no—
64#815—34no—
65C23×C32—41no—
66C22×C3×D3—42no—
67C22×C3×D3—44no—
68C22×C3×D3—82no—
69C22×C3×D3—82no—
70C2×C3⋊Dic3—42no—
71C2×C3⋊Dic3—82no—
72C2×C3×Dic3—42no—
73C2×C3×Dic3—82no—
74C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
75C22×SD16—31no—
76C2×C4×Q8—31no—
77C2×C4×D4—91no—
78#440—94no—
79#440—94no—
80C2×#107—91no—
81C2×#103—91no—
82C2×#102—91no—
83C2×#97—91no—
84C24×C3—61no—
85C23×D3—41no—
86C2×C3×Q8—31no—
87C2×C3×Q8—32no—
88C2×C3×Q8C4×D312yes—
89C2×C3×Q8C22×D311yescharacteristic
90C2×C3×D4—32no—
91C2×C3×D4—34no—
92C2×C3×D4—61no—
93C2×C3×D4—62no—
94C22×C3×C4—31no—
95C22×C3×C4—121no—
96C22×Dic3—31no—
97C22×Dic3—64no—
98C22×Dic3—61no—
99C22×Dic3—121no—
100C22×Dic3—124no—
101C22×Dic3—241no—
102C22×Dic3S411yescharacteristic
103C2×SL(2,𝔽3)—22no—
104C2×SL(2,𝔽3)—21no—
105C2×SL(2,𝔽3)C2×Dic312yescomplemented
106C2×SL(2,𝔽3)C22×D311yescharacteristic
107GL(2,𝔽3)C2×Dic314yescomplemented
108C3×SD16—38no—
109C3×SD16—34no—
110C3×SD16—34no—
111C2×C3×C8—32no—
112C2×C3×C8—34no—
113#217—64no—
114(C2×D3)⋊C4—124no—
115C4⋊Dic3—32no—
116C4⋊Dic3—32no—
117C4⋊Dic3—61no—
118C4⋊Dic3—61no—
119C4×Dic3—32no—
120C4×Dic3—32no—
121C2×C3⋊C8—32no—
122C2×C3⋊C8—62no—
123C22×C32—42no—
124C22×C32—42no—
125C22×C32—42no—
126C22×C32—41no—
127C2×C3×D3—44no—
128C2×C3×D3—88no—
129C2×C3×D3—82no—
130C2×C3×D3—82no—
131C3⋊Dic3—82no—
132C3×Dic3—82no—
133C22×Q8D3211yescharacteristic
134C22×D4—31no—
135C23×C4—181no—
136C2×SD16—32no—
137C2×SD16—34no—
138C2×SD16—34no—
139C2×SD16—32no—
140C22×C8—31no—
141C22×C8—91no—
142C4×Q8—32no—
143C4×Q8—92no—
144C4×D4—94no—
145C4×D4—182no—
146C2×C4⋊C4—91no—
147C2×C4⋊C4—91no—
148C2×C22⋊C4—181no—
149C2×C42—91no—
150#107—94no—
151#103—92no—
152#103—92no—
153#102—94no—
154#97—94no—
155C23×C3—41no—
156C23×C3—64no—
157C23×C3—62no—
158C23×C3—81no—
159C23×C3—122no—
160C23×C3—122no—
161C23×C3C2×S411yescharacteristic
162C22×D3—42no—
163C22×D3—44no—
164C22×D3—82no—
165C22×D3—82no—
166C3×Q8—32no—
167C3×Q8—31no—
168C3×Q8—61no—
169C3×Q8C2×C4×D311yescharacteristic
170C3×D4—34no—
171C3×D4—62no—
172C3×D4—64no—
173C2×C3×C4—32no—
174C2×C3×C4—32no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—122no—
178C2×C3×C4—242no—
179C2×Dic3—22no—
180C2×Dic3—32no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—122no—
184C2×Dic3—128no—
185C2×Dic3—122no—
186C2×Dic3—242no—
187C2×Dic3—242no—
188C2×Dic3—244no—
189C2×Dic3C2×S412yes—
190SL(2,𝔽3)—21no—
191SL(2,𝔽3)C22×Dic311yescharacteristic, complemented
192C3×C8—34no—
193C3⋊C8—62no—
194C2×C32—44no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C3×D3—84no—
199C24—61no—
200C2×Q8—31no—
201C2×Q8—32no—
202C2×Q8D3×Dic312yes—
203C2×Q8C2×D3211yescharacteristic
204C2×D4—32no—
205C2×D4—34no—
206C2×D4—61no—
207C2×D4—62no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—91no—
211C22×C4—184no—
212C22×C4—181no—
213C22×C4—364no—
214SD16—38no—
215SD16—34no—
216SD16—34no—
217C2×C8—32no—
218C2×C8—34no—
219C2×C8—92no—
220C2×C8—182no—
221C4⋊C4—92no—
222C4⋊C4—92no—
223C4⋊C4—181no—
224C4⋊C4—181no—
225C22⋊C4—184no—
226C42—92no—
227C42—92no—
228C22×C3—41no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—42no—
232C22×C3—64no—
233C22×C3—81no—
234C22×C3—82no—
235C22×C3—82no—
236C22×C3—82no—
237C22×C3—128no—
238C22×C3—122no—
239C22×C3—122no—
240C22×C3C4×S412yes—
241C22×C3#98312yes—
242C22×C3#98612yes—
243C22×C3C22×S411yescharacteristic
244C2×D3—44no—
245C2×D3—82no—
246C2×D3—82no—
247C2×D3—88no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—32no—
251C3×C4—242no—
252Dic3—22no—
253Dic3—62no—
254Dic3—122no—
255Dic3—242no—
256Dic3—244no—
257C32—41noSylow
258C23—62no—
259C23—64no—
260C23—122no—
261C23—122no—
262C23D3×S411yescharacteristic
263Q8—32no—
264Q8—31no—
265Q8—61no—
266Q8C2×D3×Dic311yescharacteristic
267D4—34no—
268D4—62no—
269D4—64no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—62no—
276C2×C4—92no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—368no—
280C2×C4—362no—
281C8—34no—
282C8—182no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—44no—
287C2×C3—81no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3—84no—
291C2×C3—124no—
292C2×C3#539414yescomplemented
293C2×C3C2×C4×S411yescharacteristic
294C2×C3C2×#98311yescharacteristic
295C2×C3C2×#98611yescharacteristic
296D3—84no—
297C22—64no—
298C22—122no—
299C22—122no—
300C22—128no—
301C22#6419912yes—
302C22#6420112yes—
303C22Dic3×S412yes—
304C22C2×D3×S411yescharacteristic
305C4—31no—
306C4—31no—
307C4—32no—
308C4—62no—
309C4—182no—
310C4—362no—
311C3—41no—
312C3—81no—
313C3C2×#539411yescharacteristic, complemented
314C2—124no—
315C2#1059320414yesdirect product factor
316C2C2×#6419911yescharacteristic
317C2C2×#6420111yescharacteristic
318C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C2×D3×S4
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radicalC22×C3×GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×SL(2,𝔽3)C2semidirect
3. C2×C3⋊#859C2semidirect
4. #10593204C2direct (N × H)
8. C2×C3×GL(2,𝔽3)C4semidirect
11. Dic3×SL(2,𝔽3)C22semidirect
12. C3⋊#859C22semidirect
15. C2×Q8×Dic3D3semidirect
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
32. C3×GL(2,𝔽3)C2×C4semidirect
40. C2×GL(2,𝔽3)Dic3semidirect
49. Q8×Dic3C2×D3semidirect
74. C3×SL(2,𝔽3)C22×C4semidirect
105. C2×SL(2,𝔽3)C2×Dic3semidirect
107. GL(2,𝔽3)C2×Dic3semidirect
191. SL(2,𝔽3)C22×Dic3semidirect
292. C2×C3#5394semidirect
313. C3C2×#5394semidirect
315. C2#10593204direct (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
C2×Dic3×SL(2,𝔽3)2
C2×C3⋊#8592
#105932042
C2×#53943
C2×#51643
C2×(C2×D3)⋊Dic34