Finite Group Databasebeta

← C2×#10593205 #11922111

Subgroups — C2×#10593205 #11922111

Order 1152.

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

normal subgroups only

This group has 653 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⋊#226C211yesmaximal, characteristic, complemented
2C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593205C214yesmaximal, direct product factor
5C2×#5395—31nomaximal
6C2×#5166—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊#226—22no—
9C2×C3⋊#226C2212yescomplemented
10C2×C3×GL(2,𝔽3)—22no—
11C2×C3×GL(2,𝔽3)C2212yes—
12C2×(C2×D3)⋊Dic3—41nomaximal
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C22×GL(2,𝔽3)D311yescharacteristic
15C22×#226—31no—
16C22×#226—61no—
17C22×C3×SD16—31no—
18C2×Q8×Dic3D311yescharacteristic, complemented
19C22×#216—31no—
20C2×#936—31no—
21C2×C4×SL(2,𝔽3)—31no—
22#5395—34no—
23C2×#832—31no—
24#5166—34no—
25#5166—34no—
26C2×#816—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⋊#226—22no—
34C3×GL(2,𝔽3)—22no—
35(C2×D3)⋊Dic3—44no—
36C2×#464—91noSylow
37C22×C3×Q8C2×D311yescharacteristic
38C22×C3×D4—31no—
39C22×Dic6—31no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)C2×D311yescharacteristic
42C2×GL(2,𝔽3)—22no—
43C2×GL(2,𝔽3)C2×D312yes—
44C2×#226—32no—
45C2×#226—62no—
46C2×#226—64no—
47C2×#226—62no—
48C2×C3×SD16—32no—
49C2×C3×SD16—32no—
50C2×C3×SD16—62no—
51C2×C3×SD16—62no—
52C22×C3×C8—31no—
53C2×#217—61no—
54C2×#217—61no—
55Q8×Dic3—32no—
56Q8×Dic3C2×D312yescomplemented
57C2×#216—32no—
58C2×#216—32no—
59C2×#216—62no—
60C2×#216—62no—
61#936—34no—
62#936—62no—
63C2×(C2×D3)⋊C4—121no—
64C2×C4⋊Dic3—31no—
65C2×C4×Dic3—31no—
66C22×C3⋊C8—31no—
67C4×SL(2,𝔽3)—32no—
68#832—34no—
69#816—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—82no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—82no—
80C3×SL(2,𝔽3)C2×D411yescharacteristic
81C22×Q16—91no—
82C22×SD16—31no—
83C2×#124—91no—
84C2×C4×Q8—31no—
85#464—94no—
86#464—94no—
87C2×C4⋊C8—91no—
88C2×#103—91no—
89C2×#102—91no—
90C24×C3—61no—
91C23×D3—41no—
92C2×C3×Q8—31no—
93C2×C3×Q8—32no—
94C2×C3×Q8D1212yes—
95C2×C3×Q8C22×D311yescharacteristic
96C2×C3×D4—32no—
97C2×C3×D4—62no—
98C2×C3×D4—62no—
99C2×C3×D4—61no—
100C22×C3×C4—31no—
101C22×C3×C4—121no—
102C22×Dic3—31no—
103C22×Dic3—61no—
104C22×Dic3—121no—
105C22×Dic3—241no—
106C22×Dic3S411yescharacteristic
107C2×Dic6—32no—
108C2×Dic6—62no—
109C2×Dic6—62no—
110C2×Dic6—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)—22no—
116#226—64no—
117#226—62no—
118C3×SD16—64no—
119C3×SD16—62no—
120C3×SD16—62no—
121C2×C3×C8—32no—
122C2×C3×C8—62no—
123#217—64no—
124#217—64no—
125#216—64no—
126#216—62no—
127#216—62no—
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—91no—
146C22×Q8D3211yescharacteristic
147C22×D4—31no—
148C2×Q16—92no—
149C2×Q16—92no—
150C2×Q16—182no—
151C2×Q16—182no—
152C2×SD16—32no—
153C2×SD16—32no—
154C2×SD16—62no—
155C2×SD16—62no—
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—62no—
171C23×C3—81no—
172C23×C3—122no—
173C23×C3—122no—
174C23×C3—122no—
175C23×C3C2×S411yescharacteristic
176C22×D3—42no—
177C22×D3—44no—
178C22×D3—82no—
179C22×D3—82no—
180C3×Q8—32no—
181C3×Q8—31no—
182C3×Q8—61no—
183C3×Q8C2×D1211yescharacteristic
184C3×D4—62no—
185C3×D4—62no—
186C3×D4—122no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—122no—
192C2×C3×C4—242no—
193C2×Dic3—22no—
194C2×Dic3—32no—
195C2×Dic3—62no—
196C2×Dic3—62no—
197C2×Dic3—122no—
198C2×Dic3—122no—
199C2×Dic3—242no—
200C2×Dic3—242no—
201C2×Dic3—244no—
202C2×Dic3C2×S412yes—
203Dic6—62no—
204Dic6—62no—
205Dic6—122no—
206SL(2,𝔽3)—21no—
207SL(2,𝔽3)C2×#6711yescharacteristic
208C3×C8—62no—
209C3⋊C8—62no—
210C2×C32—44no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—41no—
214C3×D3—84no—
215C24—61no—
216C2×Q8—31no—
217C2×Q8—32no—
218C2×Q8—92no—
219C2×Q8—181no—
220C2×Q8—182no—
221C2×Q8—182no—
222C2×Q8#62612yes—
223C2×Q8C2×D3211yescharacteristic
224C2×D4—32no—
225C2×D4—62no—
226C2×D4—62no—
227C2×D4—61no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—91no—
231C22×C4—181no—
232Q16—184no—
233Q16—182no—
234Q16—182no—
235SD16—64no—
236SD16—62no—
237SD16—62no—
238C2×C8—32no—
239C2×C8—62no—
240C2×C8—92no—
241C2×C8—182no—
242C4⋊C4—92no—
243C4⋊C4—181no—
244C22⋊C4—184no—
245C22⋊C4—184no—
246C42—92no—
247C42—92no—
248C22×C3—41no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—81no—
253C22×C3—82no—
254C22×C3—82no—
255C22×C3—82no—
256C22×C3—122no—
257C22×C3—122no—
258C22×C3—122no—
259C22×C3—244no—
260C22×C3C4⋊S412yes—
261C22×C3#98312yes—
262C22×C3#98512yes—
263C22×C3C22×S411yescharacteristic
264C2×D3—44no—
265C2×D3—82no—
266C2×D3—82no—
267C2×D3—88no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—32no—
271C3×C4—242no—
272Dic3—22no—
273Dic3—62no—
274Dic3—122no—
275Dic3—242no—
276Dic3—244no—
277C32—41noSylow
278C23—62no—
279C23—122no—
280C23—122no—
281C23—122no—
282C23D3×S411yescharacteristic
283Q8—32no—
284Q8—31no—
285Q8—61no—
286Q8—182no—
287Q8—182no—
288Q8—362no—
289Q8C2×#62611yescharacteristic
290D4—62no—
291D4—62no—
292D4—122no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—62no—
299C2×C4—92no—
300C2×C4—182no—
301C2×C4—182no—
302C2×C4—362no—
303C8—62no—
304C8—182no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—44no—
309C2×C3—81no—
310C2×C3—81no—
311C2×C3—81no—
312C2×C3—84no—
313C2×C3—242no—
314C2×C3#539514yescomplemented
315C2×C3C2×C4⋊S411yescharacteristic
316C2×C3C2×#98311yescharacteristic
317C2×C3C2×#98511yescharacteristic
318D3—84no—
319C22—122no—
320C22—122no—
321C22—122no—
322C22—244no—
323C22#6419712yes—
324C22#6420112yes—
325C22#6420712yes—
326C22C2×D3×S411yescharacteristic
327C4—31no—
328C4—31no—
329C4—32no—
330C4—62no—
331C4—182no—
332C4—362no—
333C3—41no—
334C3—81no—
335C3C2×#539511yescharacteristic, complemented
336C2—242no—
337C2#1059320514yesdirect product factor
338C2C2×#6419711yescharacteristic
339C2C2×#6420111yescharacteristic
340C2C2×#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
1. C22×C3⋊#226C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. #10593205C2direct (N × H)
9. C2×C3⋊#226C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
18. C2×Q8×Dic3D3semidirect
56. Q8×Dic3C2×D3semidirect
314. C2×C3#5395semidirect
335. C3C2×#5395semidirect
337. C2#10593205direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#2262
C22×C3×GL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
#105932052
C2×#53953
C2×#51663
C2×(C2×D3)⋊Dic34