Finite Group Databasebeta

← #11922878

Subgroups — #11922878

Order 1152.

582 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593655C211yesmaximal, characteristic
3#10593653C211yesmaximal, characteristic, complemented
4#10593629C211yesmaximal, characteristic, complemented
5C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#10593623C211yesmaximal, characteristic, complemented
7C4×C3⋊#226C211yesmaximal, characteristic, complemented
8#85329—33nomaximal
9#82443—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13C4×#638—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C2×C3×C4×Q8D311yescharacteristic, complemented
19#5479—31no—
20C2×C4×SL(2,𝔽3)D313yescomplemented
21#5421—33no—
22#5419—33no—
23#5395—33no—
24C4×GL(2,𝔽3)—33no—
25#5389—33no—
26C4×#226—33no—
27#5228—31no—
28#5227—31no—
29C4×#216—31no—
30C4×#216—31no—
31#5028—31no—
32#5028—31no—
33C4×#215—31no—
34C4×#215—31no—
35#5022—31no—
36#5022—31no—
37#4971—31no—
38#4967—31no—
39#4925—31no—
40C22×C32×C4—41no—
41C2×#638—41no—
42C2×C4×C3⋊D3—41no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45C3⋊GL(2,𝔽3)—21no—
46C3⋊#226—21no—
47#3757—41no—
48#3752—41no—
49#3750—41no—
50C4×C3⋊Dic3—41no—
51#3108—91noSylow
52C22×C3×Q8C2×D311yescharacteristic
53C22×SL(2,𝔽3)C2×D313yes—
54C2×GL(2,𝔽3)—33no—
55C2×#226—33no—
56C3×C4×Q8—31no—
57C3×C4×Q8—31no—
58C3×C4×Q8C2×D311yescharacteristic
59C3×C4×Q8C2×D311yescharacteristic
60C2×C3×C4⋊C4—31no—
61C2×C3×C42—31no—
62C2×#216—31no—
63C2×#216—31no—
64C2×#215—31no—
65C2×#215—31no—
66#930—31no—
67#929—61no—
68C4×#67—61no—
69C4×#67—123no—
70#924—31no—
71#876—61no—
72#875—31no—
73C4×D12—31no—
74#872—61no—
75#870—31no—
76C4×Dic6—31no—
77C4×SL(2,𝔽3)C2×D313yes—
78C4×SL(2,𝔽3)C2×D313yes—
79#859—33no—
80#859—33no—
81#835—31no—
82#835—31no—
83#835—31no—
84#835—31no—
85#830—31no—
86#830—31no—
87#810—31no—
88#810—31no—
89#809—31no—
90#809—31no—
91#808—31no—
92#808—31no—
93#807—31no—
94#807—31no—
95C4⋊(C3⋊C8)—31no—
96C4⋊(C3⋊C8)—31no—
97C4×C3⋊C8—31no—
98C4×C3⋊C8—31no—
99C23×C32—41no—
100C22×C3⋊D3—41no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C32×C4—82no—
104#638—44no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107C2×C3⋊Dic3—41no—
108C4×C3⋊D3—82no—
109C3×SL(2,𝔽3)#4111yescharacteristic
110#529—91no—
111C2×C4×Q8C3⋊D311yescharacteristic, complemented
112#483—91no—
113#480—91no—
114#479—91no—
115#479—91no—
116#464—91no—
117#464—91no—
118#451—91no—
119#448—91no—
120C4×Q16—91no—
121C4×Q16—91no—
122C4×SD16—91no—
123C4×SD16—91no—
124#431—91no—
125C2×C3×Q8—21no—
126C2×C3×Q8—31no—
127C2×C3×Q8—61no—
128C2×C3×Q8C22×D311yescharacteristic
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—43no—
132C22×C3×C4S411yescharacteristic
133C2×#67—61no—
134C2×#67—123no—
135C2×D12—31no—
136C2×C4×D3—61no—
137C2×C4×D3—123no—
138C2×Dic6—31no—
139C2×SL(2,𝔽3)—23no—
140C2×SL(2,𝔽3)C22×D313yes—
141GL(2,𝔽3)—63no—
142#226—63no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—61no—
146C3×C42—31no—
147C3×C42—31no—
148C3×C42—31no—
149C3×C42—31no—
150#217—61no—
151#217—123no—
152#216—61no—
153#216—62no—
154#216—61no—
155#215—61no—
156#215—62no—
157#215—61no—
158(C2×D3)⋊C4—61no—
159(C2×D3)⋊C4—61no—
160(C2×D3)⋊C4—61no—
161(C2×D3)⋊C4—61no—
162(C2×D3)⋊C4—123no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—61no—
168Dic3⋊C4—61no—
169Dic3⋊C4—123no—
170C4×Dic3—61no—
171C4×Dic3—123no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C22×C32—42no—
177C22×C32—41no—
178C22×C32—81no—
179C22×C32—81no—
180C2×C3⋊D3—42no—
181C2×C3⋊D3—81no—
182C2×C3⋊D3—81no—
183C32×C4—42no—
184C32×C4—81no—
185C3⋊Dic3—42no—
186C3⋊Dic3—81no—
187C3⋊Dic3—81no—
188C22×Q8C2×C3⋊D311yescharacteristic
189C2×Q16—91no—
190C2×Q16—91no—
191C2×SD16—91no—
192C2×SD16—91no—
193#126—181no—
194#125—91no—
195#124—91no—
196#123—181no—
197C22⋊Q8—91no—
198#121—91no—
199C4×Q8—31no—
200C4×Q8—31no—
201C4×Q8—91no—
202C4×Q8C2×C3⋊D311yescharacteristic
203C4×Q8C2×C3⋊D311yescharacteristic
204C4×D4—91no—
205C4×D4—181no—
206#117—181no—
207C2×C4⋊C4—31no—
208C2×C42—31no—
209#107—91no—
210#107—91no—
211#106—91no—
212#106—91no—
213C4⋊C8—91no—
214C4⋊C8—91no—
215#103—91no—
216#103—91no—
217#103—91no—
218#103—91no—
219#103—91no—
220#103—91no—
221#102—91no—
222#102—91no—
223C22⋊C8—91no—
224C22⋊C8—91no—
225C4×C8—91no—
226C4×C8—91no—
227C23×C3—43no—
228C23×C3C2×S411yescharacteristic
229C22×D3—61no—
230C22×D3—123no—
231C3×Q8—31no—
232C3×Q8—32no—
233C3×Q8—61no—
234C3×Q8#23511yescharacteristic
235C2×C3×C4—22no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—43no—
241C2×C3×C4—43no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—62no—
245C2×C3×C4—86no—
246C2×C3×C4C2×S411yescharacteristic
247C2×C3×C4C2×S411yescharacteristic
248#67—1212no—
249#67—122no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—123no—
254C2×Dic3—123no—
255C2×Dic3—123no—
256D12—61no—
257D12—61no—
258C4×D3—122no—
259C4×D3—246no—
260Dic6—61no—
261Dic6—61no—
262SL(2,𝔽3)#23513yes—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C3⋊C8—61no—
266C3⋊C8—61no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—82no—
271C3⋊D3—82no—
272C2×Q8—21no—
273C2×Q8—31no—
274C2×Q8—61no—
275C2×Q8—91no—
276C2×Q8C22×C3⋊D311yescharacteristic
277C2×D4—91no—
278C2×D4—181no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—181no—
282C22×C4C3⋊S411yescharacteristic
283Q16—181no—
284Q16—182no—
285Q16—181no—
286SD16—181no—
287SD16—182no—
288SD16—181no—
289C2×C8—91no—
290C2×C8—91no—
291C2×C8—91no—
292C2×C8—91no—
293C4⋊C4—31no—
294C4⋊C4—31no—
295C4⋊C4—61no—
296C4⋊C4—91no—
297C4⋊C4—91no—
298C4⋊C4—181no—
299C4⋊C4—181no—
300C4⋊C4—181no—
301C4⋊C4—181no—
302C22⋊C4—181no—
303C22⋊C4—181no—
304C22⋊C4—181no—
305C22⋊C4—181no—
306C22⋊C4—181no—
307C42—31no—
308C42—31no—
309C42—31no—
310C42—31no—
311C42—181no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—46no—
315C22×C3—43no—
316C22×C3—83no—
317C22×C3—83no—
318C22×C3#98512yes—
319C22×C3C22×S411yescharacteristic
320C2×D3—126no—
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—243no—
325C2×D3—243no—
326C3×C4—21no—
327C3×C4—31no—
328C3×C4—31no—
329C3×C4—46no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—61no—
333C3×C4—83no—
334C3×C4#98312yes—
335Dic3—126no—
336Dic3—121no—
337Dic3—121no—
338Dic3—121no—
339Dic3—243no—
340Dic3—243no—
341C32—41noSylow
342C23—181no—
343C23C2×C3⋊S411yescharacteristic
344Q8—31no—
345Q8—32no—
346Q8—61no—
347Q8—181no—
348Q8—181no—
349Q8#382811yescharacteristic
350D4—181no—
351D4—181no—
352D4—362no—
353C2×C4—22no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—62no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4—362no—
365C2×C4C2×C3⋊S411yescharacteristic
366C2×C4C2×C3⋊S411yescharacteristic
367C8—181no—
368C8—181no—
369C8—181no—
370C8—181no—
371C2×C3—22no—
372C2×C3—43no—
373C2×C3—43no—
374C2×C3—43no—
375C2×C3—86no—
376C2×C3#591211yescharacteristic
377C2×C3C2×#98311yescharacteristic
378C2×C3C2×#98511yescharacteristic
379D3—246no—
380D3—241no—
381C22—21no—
382C22—21no—
383C22—361no—
384C22—361no—
385C22—361no—
386C22#6426612yes—
387C22C22×C3⋊S411yescharacteristic
388C4—21no—
389C4—31no—
390C4—31no—
391C4—61no—
392C4—61no—
393C4—61no—
394C4—361no—
395C4—361no—
396C4—361no—
397C4#6426412yes—
398C3—43no—
399C3#8532911yescharacteristic, complemented
400C2—22no—
401C2—721no—
402C2#1059657511yescharacteristic
403C2C2×#6426411yescharacteristic
404C2C2×#6426611yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
prime radical#10593655C2
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. #10593653C2semidirect
4. #10593629C2semidirect
5. C4×C3⋊GL(2,𝔽3)C2semidirect
6. #10593623C2semidirect
7. C4×C3⋊#226C2semidirect
18. C2×C3×C4×Q8D3semidirect
20. C2×C4×SL(2,𝔽3)D3semidirect
111. C2×C4×Q8C3⋊D3semidirect
399. C3#85329semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#105936552
#105936532
#105936292
C4×C3⋊GL(2,𝔽3)2
#105936232
C4×C3⋊#2262
#853293
#824433
C4×#6384