Finite Group Databasebeta

← D4×C3⋊#226 #11922939

Subgroups — D4×C3⋊#226 #11922939

Order 1152.

698 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 87 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⋊#226C212yesmaximal, complemented
2C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic
3#10593653C212yesmaximal, complemented
4C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
5C4×C3⋊#226C211yesmaximal, characteristic, complemented
6D4×#226—33nomaximal
7D4×#216—31nomaximal
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×C3⋊#226—24no—
10C2×C3⋊#226C2212yes—
11C2×C3⋊#226C2211yescharacteristic
12D4×C3⋊Dic3—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3⋊#859C2211yescharacteristic
15C22×#226—36no—
16C3×D4×Q8D311yescharacteristic
17C22×#216—32no—
18D4×Dic6—31no—
19D4×SL(2,𝔽3)D313yes—
20#5419—36no—
21C4⋊#226—33no—
22C4×#226—33no—
23#5228—32no—
24#5092—31no—
25#5090—31no—
26#5051—32no—
27#5050—32no—
28#5031—31no—
29C4×#216—31no—
30D4×C3⋊C8—31no—
31C2×C32×D4—41no—
32C22×C3⋊Dic3—42no—
33C2×C3×SL(2,𝔽3)—22no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊#226—22no—
36C3⋊#226D412yesdirect product factor
37#3757—42no—
38C4⋊(C3⋊Dic3)—41no—
39C4×C3⋊Dic3—41no—
40D4×Q16—91noSylow
41C22×C3×Q8C2×D312yes—
42C22×Dic6—32no—
43C22×SL(2,𝔽3)C2×D316yes—
44C2×#226—33no—
45C2×#226—36no—
46C2×#226—612no—
47C3×C4⋊Q8—31no—
48C3×C22⋊Q8—32no—
49C3×C4×Q8C2×D311yescharacteristic
50C3×C4×D4—31no—
51C2×#216—31no—
52C2×#216—32no—
53C2×#216—32no—
54C2×#216—32no—
55C2×#216—32no—
56C2×#216—64no—
57C2×#216—64no—
58D4×Dic3—61no—
59D4×Dic3—123no—
60#924—32no—
61C22×C3⋊C8—32no—
62#888—61no—
63#878—62no—
64#869—31no—
65C4×Dic6—31no—
66C4×SL(2,𝔽3)C2×D313yes—
67#859—33no—
68#835—32no—
69#835—31no—
70#830—32no—
71#810—31no—
72#810—32no—
73#807—31no—
74C4⋊(C3⋊C8)—31no—
75C4×C3⋊C8—31no—
76C23×C32—42no—
77C32×D4—44no—
78C2×C32×C4—41no—
79C2×C3⋊Dic3—41no—
80C2×C3⋊Dic3—42no—
81C2×C3⋊Dic3—41no—
82C2×C3⋊Dic3—84no—
83C3×SL(2,𝔽3)C2×D411yescharacteristic
84C22×Q16—92no—
85D4×Q8—91no—
86D4×Q8C3⋊D311yescharacteristic
87#494—91no—
88#467—92no—
89#462—91no—
90#462—91no—
91#451—92no—
92#451—92no—
93C4×Q16—91no—
94C8×D4—91no—
95C2×C3×Q8—22no—
96C2×C3×Q8—32no—
97C2×C3×Q8—62no—
98C2×C3×Q8C22×D311yescharacteristic
99C2×C3×D4—43no—
100C2×C3×D4S411yescharacteristic
101C22×C3×C4—32no—
102C22×Dic3—62no—
103C22×Dic3—126no—
104C2×Dic6—31no—
105C2×Dic6—32no—
106C2×Dic6—64no—
107C2×Dic6—62no—
108C2×Dic6—122no—
109C2×SL(2,𝔽3)—26no—
110C2×SL(2,𝔽3)C22×D313yes—
111#226—36no—
112#226—66no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—31no—
115C3×C4⋊C4—61no—
116C3×C22⋊C4—32no—
117C3×C42—31no—
118#217—62no—
119#217—126no—
120#216—32no—
121#216—32no—
122#216—64no—
123#216—64no—
124#216—62no—
125#216—62no—
126#216—122no—
127C4⋊Dic3—31no—
128C4⋊Dic3—61no—
129C4⋊Dic3—61no—
130C4⋊Dic3—123no—
131Dic3⋊C4—61no—
132Dic3⋊C4—62no—
133C4×Dic3—61no—
134C4×Dic3—123no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—32no—
138C2×C3⋊C8—64no—
139C22×C32—44no—
140C22×C32—41no—
141C22×C32—82no—
142C22×C32—82no—
143C32×C4—42no—
144C3⋊Dic3—42no—
145C3⋊Dic3—81no—
146C3⋊Dic3—82no—
147C22×Q8—92no—
148C22×Q8C2×C3⋊D312yes—
149C2×Q16—91no—
150C2×Q16—92no—
151C2×Q16—92no—
152C2×Q16—92no—
153C2×Q16—92no—
154C2×Q16—184no—
155C2×Q16—184no—
156C22×C8—92no—
157C4⋊Q8—31no—
158C4⋊Q8—91no—
159C4⋊Q8—181no—
160C22⋊Q8—32no—
161C22⋊Q8—92no—
162C22⋊Q8—182no—
163C4×Q8—91no—
164C4×Q8C2×C3⋊D311yescharacteristic
165C4×D4—31no—
166C4×D4—181no—
167#107—91no—
168C4⋊C8—91no—
169#103—91no—
170#103—92no—
171#103—92no—
172#103—91no—
173C22⋊C8—92no—
174C4×C8—91no—
175C23×C3—46no—
176C23×C3C2×S412yes—
177C3×Q8—31no—
178C3×Q8—62no—
179C3×Q8—62no—
180C3×Q8D3×D411yescharacteristic
181C3×D4—412no—
182C3×D4#22614yescomplemented
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—43no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4C2×S411yescharacteristic
190C2×Dic3—61no—
191C2×Dic3—61no—
192C2×Dic3—62no—
193C2×Dic3—123no—
194C2×Dic3—123no—
195C2×Dic3—126no—
196C2×Dic3—124no—
197C2×Dic3—2412no—
198Dic6—32no—
199Dic6—61no—
200Dic6—62no—
201Dic6—62no—
202Dic6—124no—
203SL(2,𝔽3)D3×D413yes—
204C3⋊C8—32no—
205C3⋊C8—62no—
206C3⋊C8—61no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—84no—
211C2×Q8—22no—
212C2×Q8—32no—
213C2×Q8—62no—
214C2×Q8—91no—
215C2×Q8—92no—
216C2×Q8—182no—
217C2×Q8—184no—
218C2×Q8—362no—
219C2×Q8C22×C3⋊D311yescharacteristic
220C2×D4C3⋊S411yescharacteristic
221C22×C4—32no—
222C22×C4—182no—
223Q16—92no—
224Q16—92no—
225Q16—184no—
226Q16—182no—
227Q16—184no—
228Q16—182no—
229Q16—362no—
230C2×C8—91no—
231C2×C8—91no—
232C2×C8—92no—
233C2×C8—184no—
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—61no—
237C4⋊C4—91no—
238C4⋊C4—181no—
239C4⋊C4—182no—
240C4⋊C4—181no—
241C4⋊C4—181no—
242C22⋊C4—32no—
243C22⋊C4—182no—
244C42—31no—
245C42—181no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—412no—
249C22×C3—43no—
250C22×C3—86no—
251C22×C3—86no—
252C22×C3C2×#22614yescomplemented
253C22×C3C22×S411yescharacteristic
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—46no—
257C3×C4—62no—
258C3×C4—61no—
259C3×C4C2×#22612yescomplemented
260Dic3—62no—
261Dic3—126no—
262Dic3—122no—
263Dic3—121no—
264Dic3—246no—
265Dic3—243no—
266C32—41noSylow
267C23C2×C3⋊S412yes—
268Q8—31no—
269Q8—62no—
270Q8—62no—
271Q8—92no—
272Q8—181no—
273Q8—182no—
274Q8—182no—
275Q8—364no—
276Q8D4×C3⋊D311yescharacteristic
277D4C3⋊#22614yesdirect product factor
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C4—182no—
286C2×C4—364no—
287C2×C4C2×C3⋊S411yescharacteristic
288C8—92no—
289C8—182no—
290C8—181no—
291C2×C3—24no—
292C2×C3—43no—
293C2×C3—43no—
294C2×C3—43no—
295C2×C3—812no—
296C2×C3D4×S411yescharacteristic
297C2×C3C22×#22611yescharacteristic
298C2×C3#592711yescharacteristic
299C22—22no—
300C22—22no—
301C22C2×C3⋊#22614yescomplemented
302C22C22×C3⋊S411yescharacteristic
303C4—31no—
304C4—31no—
305C4—62no—
306C4—61no—
307C4—182no—
308C4—362no—
309C4—361no—
310C4C2×C3⋊#22612yescomplemented
311C3—43no—
312C3D4×#22611yescharacteristic, complemented
313C2—24no—
314C2D4×C3⋊S411yescharacteristic
315C2C22×C3⋊#22611yescharacteristic
316C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×D4×Q8D3
socleC22×C3C22×S4
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊#226C2semidirect
3. #10593653C2semidirect
4. C4⋊(C3⋊#226)C2semidirect
5. C4×C3⋊#226C2semidirect
36. C3⋊#226D4direct (N × H)
182. C3×D4#226semidirect
252. C22×C3C2×#226semidirect
259. C3×C4C2×#226semidirect
277. D4C3⋊#226direct (N × H)
301. C22C2×C3⋊#226semidirect
310. C4C2×C3⋊#226semidirect
312. C3D4×#226semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#2262
C3×D4×SL(2,𝔽3)2
#105936532
C4⋊(C3⋊#226)2
C4×C3⋊#2262
D4×#2263
D4×#2163
D4×C3⋊Dic34