Finite Group Databasebeta

← #11922899

Subgroups — #11922899

Order 1152.

644 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 89 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
2C3×#519⋊C3C211yesmaximal, characteristic
3#10593654C211yesmaximal, characteristic
4#10593622C212yesmaximal, complemented
5C4×C3⋊#226C212yesmaximal, complemented
6#85350—33nomaximal
7#82549—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊#226C414yescomplemented
10C2×C3⋊#226C2211yescharacteristic
11C2×C3⋊#226C2211yescharacteristic
12#64086—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yes—
14C3⋊#859C2212yes—
15C22×#226—33no—
16C3×#519D311yescharacteristic
17C22×#216—31no—
18#5516—31no—
19#519⋊C3D313yes—
20#5420—33no—
21#5388—36no—
22C4×#226—36no—
23#5225—31no—
24#5030—32no—
25#5030—32no—
26C4×#216—32no—
27C4×#216—32no—
28#5004—31no—
29#4998—31no—
30#4997—31no—
31C22×C3⋊Dic3—41no—
32C2×C3×SL(2,𝔽3)C2×C412yes—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊#226C2×C412yes—
35C3⋊#226C2×C412yes—
36C32×C22⋊C4—41no—
37#3757—41no—
38#3750—42no—
39C4×C3⋊Dic3—42no—
40#2950—91noSylow
41C22×C3×Q8C2×D311yescharacteristic
42C22×Dic6—31no—
43C22×SL(2,𝔽3)C2×D313yes—
44C2×#226—312no—
45C2×#226—33no—
46C2×#226—33no—
47C3×C4×Q8—32no—
48C3×C4×Q8C2×D312yes—
49C3×#117—31no—
50C3×#117—31no—
51C2×#216—31no—
52C2×#216—31no—
53C2×#216—34no—
54C2×#216—34no—
55C2×#216—31no—
56C2×#216—31no—
57#926—31no—
58C2×#208—31no—
59C22×C3⋊C8—31no—
60#887—62no—
61#877—61no—
62#877—61no—
63#877—123no—
64C4×Dic6—32no—
65C4×Dic6—32no—
66C4×SL(2,𝔽3)C2×D316yes—
67#859—36no—
68#835—32no—
69#835—32no—
70#810—32no—
71#810—32no—
72#808—32no—
73#807—32no—
74#803—32no—
75C4×C3⋊C8—32no—
76C23×C32—41no—
77C2×C32×C4—42no—
78C2×C3⋊Dic3—44no—
79C2×C3⋊Dic3—41no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3—42no—
82C3×SL(2,𝔽3)C22×C411yescharacteristic
83C22×Q16—91no—
84#519—91no—
85#519C3⋊D311yescharacteristic
86#441—92no—
87#441—92no—
88C4×Q16—92no—
89C4×Q16—92no—
90#427—91no—
91#419—91no—
92#419—91no—
93#405—91no—
94C2×C3×Q8—31no—
95C2×C3×Q8—32no—
96C2×C3×Q8C4×D312yes—
97C2×C3×Q8C22×D311yescharacteristic
98C22×C3×C4—31no—
99C22×Dic3—61no—
100C22×Dic3—123no—
101C2×Dic6—31no—
102C2×Dic6—31no—
103C2×Dic6—34no—
104C2×Dic6—61no—
105C2×Dic6—62no—
106C2×SL(2,𝔽3)C4×D316yes—
107C2×SL(2,𝔽3)C22×D313yes—
108#226—36no—
109#226—36no—
110C3×C4⋊C4—32no—
111C3×C4⋊C4—32no—
112C3×C22⋊C4—31no—
113C3×C22⋊C4—43no—
114C3×C22⋊C4S411yescharacteristic
115C3×C42—32no—
116C3×C42—61no—
117#217—61no—
118#217—123no—
119#216—34no—
120#216—34no—
121#216—32no—
122#216—32no—
123#216—32no—
124#216—32no—
125C4⋊Dic3—32no—
126Dic3⋊C4—62no—
127Dic3⋊C4—62no—
128Dic3⋊C4—126no—
129C4×Dic3—62no—
130C4×Dic3—61no—
131C4×Dic3—62no—
132C4×Dic3—126no—
133#208—62no—
134C2×C3⋊C8—31no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—34no—
137C2×C3⋊C8—32no—
138C22×C32—42no—
139C22×C32—41no—
140C22×C32—82no—
141C32×C4—82no—
142C3⋊Dic3—42no—
143C3⋊Dic3—42no—
144C3⋊Dic3—82no—
145C22×Q8—91no—
146C22×Q8C2×C3⋊D311yescharacteristic
147C2×Q16—94no—
148C2×Q16—91no—
149C2×Q16—91no—
150C2×Q16—94no—
151C2×Q16—91no—
152C2×Q16—91no—
153C2×M16—91no—
154C22×C8—91no—
155C4×Q8—32no—
156C4×Q8—92no—
157C4×Q8—92no—
158C4×Q8—182no—
159C4×Q8C2×C3⋊D312yes—
160#117—31no—
161#117—31no—
162#117—91no—
163#117—181no—
164#117—181no—
165#107—92no—
166#106—92no—
167#103—92no—
168#103—92no—
169#103—92no—
170#103—92no—
171#97—92no—
172C4×C8—92no—
173C23×C3—43no—
174C23×C3C2×S411yescharacteristic
175C3×Q8—32no—
176C3×Q8—31no—
177C3×Q8—61no—
178C3×Q8C2×C4×D311yescharacteristic
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—46no—
185C2×C3×C4C2×S412yes—
186C2×Dic3—64no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—62no—
190C2×Dic3—123no—
191C2×Dic3—123no—
192C2×Dic3—1212no—
193C2×Dic3—126no—
194Dic6—32no—
195Dic6—32no—
196Dic6—61no—
197Dic6—61no—
198Dic6—64no—
199SL(2,𝔽3)C2×C4×D313yes—
200C3⋊C8—32no—
201C3⋊C8—32no—
202C3⋊C8—62no—
203C2×C32—41no—
204C2×C32—42no—
205C2×C32—82no—
206C2×Q8—31no—
207C2×Q8—32no—
208C2×Q8—91no—
209C2×Q8—91no—
210C2×Q8—94no—
211C2×Q8—181no—
212C2×Q8—182no—
213C2×Q8C4×C3⋊D312yes—
214C2×Q8C22×C3⋊D311yescharacteristic
215C22×C4—31no—
216C22×C4—181no—
217Q16—94no—
218Q16—94no—
219Q16—92no—
220Q16—92no—
221Q16—92no—
222Q16—92no—
223M16—182no—
224C2×C8—91no—
225C2×C8—91no—
226C2×C8—94no—
227C2×C8—92no—
228C4⋊C4—32no—
229C4⋊C4—32no—
230C4⋊C4—92no—
231C4⋊C4—182no—
232C4⋊C4—182no—
233C22⋊C4—31no—
234C22⋊C4—181no—
235C22⋊C4C3⋊S411yescharacteristic
236C42—32no—
237C42—61no—
238C42—181no—
239C42—182no—
240C42—182no—
241C22×C3—22no—
242C22×C3—46no—
243C22×C3—43no—
244C22×C3—86no—
245C22×C3C4×S412yes—
246C22×C3C22×S411yescharacteristic
247C3×C4—22no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—32no—
251C3×C4—62no—
252C3×C4—86no—
253Dic3—62no—
254Dic3—62no—
255Dic3—126no—
256Dic3—126no—
257Dic3—122no—
258Dic3—246no—
259C32—41noSylow
260C23C2×C3⋊S411yescharacteristic
261Q8—32no—
262Q8—31no—
263Q8—61no—
264Q8—92no—
265Q8—92no—
266Q8—181no—
267Q8—181no—
268Q8—184no—
269Q8C2×C4×C3⋊D311yescharacteristic
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—184no—
276C2×C4—181no—
277C2×C4—181no—
278C2×C4—182no—
279C2×C4C2×C3⋊S412yes—
280C8—92no—
281C8—92no—
282C8—182no—
283C2×C3—22no—
284C2×C3—43no—
285C2×C3—46no—
286C2×C3—86no—
287C2×C3C2×C4×S411yescharacteristic
288C2×C3#592712yes—
289C22—22no—
290C22C4×C3⋊S412yes—
291C22C22×C3⋊S411yescharacteristic
292C4—22no—
293C4—31no—
294C4—31no—
295C4—32no—
296C4—62no—
297C4—182no—
298C4—182no—
299C4—362no—
300C3—43no—
301C3#8535011yescharacteristic, complemented
302C2—22no—
303C2C2×C4×C3⋊S411yescharacteristic
304C2#1059659012yes—
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC3×#519D3
socleC22×C3C22×S4
prime radicalC22×C3×SL(2,𝔽3)C22
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC22⋊C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #10593622C2semidirect
5. C4×C3⋊#226C2semidirect
9. C2×C3⋊#226C4semidirect
301. C3#85350semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#2262
C3×#519⋊C32
#105936542
#105936222
C4×C3⋊#2262
#853503
#825493
#640864