Finite Group Databasebeta

← #11922937

Subgroups — #11922937

Order 1152.

548 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C3×#548⋊C3C211yesmaximal, characteristic, complemented
2#10593661C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4#10593655C211yesmaximal, characteristic, complemented
5#10593654C211yesmaximal, characteristic, complemented
6C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#10593623C211yesmaximal, characteristic, complemented
8#85387—33nomaximal
9#83540—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
13#64092—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×#548D311yescharacteristic, complemented
19#5565—31no—
20#548⋊C3D313yescomplemented
21#5427—33no—
22#5426—33no—
23#5421—33no—
24#5420—33no—
25C4×GL(2,𝔽3)—33no—
26#5389—33no—
27#5238—31no—
28#5232—31no—
29#5227—31no—
30#5225—31no—
31#5057—31no—
32#5056—31no—
33#5049—31no—
34#5043—31no—
35#5039—31no—
36#5035—31no—
37C4×#215—31no—
38#5022—31no—
39#5012—31no—
40C2×#638—41no—
41C2×C4×C3⋊D3—41no—
42C3×#41⋊C3—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45C3⋊GL(2,𝔽3)—21no—
46C32×C22⋊C4—41no—
47#3757—41no—
48#3752—41no—
49C4⋊(C3⋊Dic3)—41no—
50#3750—41no—
51#3326—91noSylow
52C2×C3×#41C2×D311yescharacteristic
53C22×C3×Q8C2×D311yescharacteristic
54C2×#41⋊C3C2×D313yes—
55C22×SL(2,𝔽3)C2×D313yes—
56C2×GL(2,𝔽3)—33no—
57C3×#124—31no—
58C3×C22⋊Q8—31no—
59C3×#121—31no—
60C3×C4×Q8C2×D311yescharacteristic
61C3×C4×D4—31no—
62C2×#215—31no—
63C2×#215—31no—
64#939—61no—
65C2×#213—31no—
66#930—31no—
67#930—31no—
68#926—31no—
69C2×C4⋊Dic3—31no—
70C2×#208—31no—
71C2×#208—31no—
72#898—61no—
73#883—61no—
74#883—123no—
75#67⋊C4—61no—
76C4×D12—31no—
77#870—31no—
78C4×SL(2,𝔽3)C2×D313yes—
79#859—33no—
80#859—33no—
81#859—33no—
82#835—31no—
83#835—31no—
84#835—31no—
85#835—31no—
86#835—31no—
87#835—31no—
88#832—31no—
89#832—31no—
90#832—31no—
91#830—31no—
92#830—31no—
93#809—31no—
94#808—31no—
95#808—31no—
96#807—31no—
97C4⋊(C3⋊C8)—31no—
98C4×C3⋊C8—31no—
99C23×C32—41no—
100C22×C3⋊D3—41no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103#638—82no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107C4×C3⋊D3—82no—
108C3×SL(2,𝔽3)#4111yescharacteristic
109#553—91no—
110#548C3⋊D311yescharacteristic, complemented
111#488—91no—
112#483—91no—
113#482—91no—
114#479—91no—
115#469—91no—
116#468—91no—
117#463—91no—
118#449—91no—
119#448—91no—
120C4×SD16—91no—
121#436—91no—
122#419—91no—
123#417—91no—
124C3×#41—21no—
125C3×#41—61no—
126C2×C3×Q8—21no—
127C2×C3×Q8—31no—
128C2×C3×Q8—61no—
129C2×C3×Q8C22×D311yescharacteristic
130C2×C3×D4—31no—
131C2×C3×D4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—31no—
134C2×#67—61no—
135C2×#67—61no—
136C2×#67—123no—
137C22×Dic3—61no—
138C2×D12—31no—
139C2×C4×D3—61no—
140C2×C4×D3—123no—
141#41⋊C3—23no—
142C2×SL(2,𝔽3)—23no—
143C2×SL(2,𝔽3)C22×D313yes—
144GL(2,𝔽3)—63no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4—43no—
149C3×C22⋊C4—61no—
150C3×C22⋊C4S411yescharacteristic
151C3×C42—31no—
152#217—61no—
153#217—123no—
154#215—61no—
155#215—61no—
156#215—121no—
157#213—121no—
158(C2×D3)⋊C4—61no—
159(C2×D3)⋊C4—61no—
160(C2×D3)⋊C4—61no—
161(C2×D3)⋊C4—123no—
162C4⋊Dic3—31no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—61no—
166C4⋊Dic3—123no—
167Dic3⋊C4—61no—
168Dic3⋊C4—61no—
169Dic3⋊C4—123no—
170C4×Dic3—61no—
171#208—121no—
172#208—121no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C22×C32—41no—
178C22×C32—81no—
179C22×C32—81no—
180C22×C32—81no—
181C2×C3⋊D3—42no—
182C2×C3⋊D3—81no—
183C2×C3⋊D3—81no—
184C32×C4—81no—
185C32×C4—81no—
186C3⋊Dic3—81no—
187C3⋊Dic3—81no—
188C3⋊Dic3—81no—
189C2×#41C2×C3⋊D311yescharacteristic
190C22×Q8C2×C3⋊D311yescharacteristic
191C2×SD16—91no—
192C2×SD16—91no—
193C2×D8—91no—
194C2×M16—91no—
195C2×M16—91no—
196#126—181no—
197#125—91no—
198#124—31no—
199#123—181no—
200C22⋊Q8—31no—
201#121—31no—
202#121—91no—
203#121—91no—
204#121—181no—
205C4×Q8C2×C3⋊D311yescharacteristic
206C4×D4—31no—
207C4×D4—91no—
208C4×D4—181no—
209#117—91no—
210C2×C4⋊C4—91no—
211#107—91no—
212#106—91no—
213#106—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—
223#102—91no—
224#102—91no—
225C22⋊C8—91no—
226C22⋊C8—91no—
227C4×C8—91no—
228C23×C3—31no—
229C23×C3—43no—
230C23×C3C2×S411yescharacteristic
231C22×D3—61no—
232C22×D3—123no—
233C3×Q8—31no—
234C3×Q8—61no—
235C3×Q8—61no—
236C3×Q8#23711yescharacteristic
237C3×D4—61no—
238C3×D4—61no—
239C3×D4—121no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—43no—
244C2×C3×C4—43no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4C2×S411yescharacteristic
250C2×C3×C4C2×S411yescharacteristic
251#67—122no—
252#67—241no—
253#67—246no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—123no—
258C2×Dic3—123no—
259C2×Dic3—123no—
260C2×Dic3—241no—
261D12—61no—
262D12—61no—
263C4×D3—241no—
264C4×D3—246no—
265SL(2,𝔽3)#23713yes—
266C3⋊C8—61no—
267C3⋊C8—61no—
268C3⋊C8—61no—
269C3⋊C8—61no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—161no—
274C3⋊D3—82no—
275#41—21no—
276#41—61no—
277C2×Q8—21no—
278C2×Q8—31no—
279C2×Q8—61no—
280C2×Q8C22×C3⋊D311yescharacteristic
281C2×D4—31no—
282C2×D4—31no—
283C2×D4—91no—
284C2×D4—181no—
285C2×D4—181no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—181no—
289C22×C4—181no—
290SD16—181no—
291SD16—181no—
292SD16—361no—
293D8—361no—
294M16—361no—
295M16—361no—
296C2×C8—91no—
297C2×C8—91no—
298C2×C8—91no—
299C2×C8—91no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—91no—
303C4⋊C4—91no—
304C4⋊C4—91no—
305C4⋊C4—181no—
306C4⋊C4—181no—
307C4⋊C4—181no—
308C22⋊C4—31no—
309C22⋊C4—61no—
310C22⋊C4—181no—
311C22⋊C4—181no—
312C22⋊C4—181no—
313C22⋊C4—181no—
314C22⋊C4C3⋊S411yescharacteristic
315C42—31no—
316C42—181no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—43no—
321C22×C3—61no—
322C22×C3—61no—
323C22×C3—61no—
324C22×C3—83no—
325C22×C3—83no—
326C22×C3—83no—
327C22×C3C22×S411yescharacteristic
328C2×D3—126no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—243no—
333C2×D3—243no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—83no—
341C3×C4—83no—
342Dic3—121no—
343Dic3—121no—
344Dic3—121no—
345Dic3—243no—
346Dic3—243no—
347Dic3—243no—
348C32—41noSylow
349C23—31no—
350C23—181no—
351C23C2×C3⋊S411yescharacteristic
352Q8—31no—
353Q8—61no—
354Q8—61no—
355Q8#383011yescharacteristic
356D4—61no—
357D4—61no—
358D4—121no—
359D4—181no—
360D4—181no—
361D4—362no—
362D4—721no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4—721no—
374C2×C4—721no—
375C2×C4C2×C3⋊S411yescharacteristic
376C2×C4C2×C3⋊S411yescharacteristic
377C8—181no—
378C8—181no—
379C8—181no—
380C8—181no—
381C2×C3—41no—
382C2×C3—43no—
383C2×C3—43no—
384C2×C3—43no—
385C2×C3—121no—
386C2×C3—163no—
387C2×C3#591411yescharacteristic
388C2×C3#592311yescharacteristic
389C2×C3#592611yescharacteristic
390D3—246no—
391D3—241no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22—361no—
399C22—361no—
400C22—361no—
401C22C22×C3⋊S411yescharacteristic
402C4—21no—
403C4—21no—
404C4—31no—
405C4—31no—
406C4—61no—
407C4—61no—
408C4—361no—
409C4—361no—
410C4—361no—
411C3—43no—
412C3#8538711yescharacteristic, complemented
413C2—41no—
414C2—121no—
415C2—721no—
416C2#1059657711yescharacteristic
417C2#1059658611yescharacteristic
418C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#548D3
socleC22×C3C22×S4
hypercenterC22⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#548⋊C3C2semidirect
2. #10593661C2semidirect
3. #10593660C2semidirect
4. #10593655C2semidirect
5. #10593654C2semidirect
6. C4×C3⋊GL(2,𝔽3)C2semidirect
7. #10593623C2semidirect
18. C3×#548D3semidirect
20. #548⋊C3D3semidirect
110. #548C3⋊D3semidirect
412. C3#85387semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#105936612
#105936602
#105936552
#105936542
C4×C3⋊GL(2,𝔽3)2
#105936232
#853873
#835403
#640924