Finite Group Databasebeta

← #11922933

Subgroups — #11922933

Order 1152.

586 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 69 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#10593658C211yesmaximal, characteristic, complemented
4#10593655C211yesmaximal, characteristic, complemented
5C2×C3⋊#859C211yesmaximal, characteristic, complemented
6C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#10593623C211yesmaximal, characteristic, complemented
8#85381—33nomaximal
9#83535—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
13#64090—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#5424—33no—
23#5421—33no—
24C2×#859—33no—
25C4×GL(2,𝔽3)—33no—
26#5389—33no—
27#5238—31no—
28#5233—31no—
29#5227—31no—
30C2×#835—31no—
31#5057—31no—
32#5056—31no—
33#5048—31no—
34#5045—31no—
35#5038—31no—
36#5034—31no—
37C4×#215—31no—
38#5022—31no—
39D4×C3⋊C8—31no—
40C2×#638—41no—
41C22×C3⋊Dic3—41no—
42C2×C4×C3⋊D3—41no—
43C3×#41⋊C3—21no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46C3⋊GL(2,𝔽3)—21no—
47C32×C22⋊C4—41no—
48#3752—41no—
49#3750—41no—
50C4×C3⋊Dic3—41no—
51#3301—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—
70C22×C3⋊C8—31no—
71C22×C3⋊C8—31no—
72#898—61no—
73#883—61no—
74#67⋊C4—61no—
75#67⋊C4—123no—
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—44no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107C2×C3⋊Dic3—82no—
108C4×C3⋊D3—82no—
109C3×SL(2,𝔽3)#4111yescharacteristic
110#553—91no—
111#548C3⋊D311yescharacteristic, complemented
112#488—91no—
113#485—91no—
114#481—91no—
115#479—91no—
116#466—91no—
117#465—91no—
118#463—91no—
119#449—91no—
120#448—91no—
121C4×SD16—91no—
122C8×D4—91no—
123#416—91no—
124C2×#103—91no—
125C3×#41—21no—
126C3×#41—61no—
127C2×C3×Q8—21no—
128C2×C3×Q8—31no—
129C2×C3×Q8—61no—
130C2×C3×Q8C22×D311yescharacteristic
131C2×C3×D4—31no—
132C2×C3×D4—31no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C2×#67—61no—
136C2×#67—61no—
137C2×#67—123no—
138C22×Dic3—61no—
139C22×Dic3—123no—
140C2×D12—31no—
141C2×C4×D3—61no—
142C2×C4×D3—123no—
143#41⋊C3—23no—
144C2×SL(2,𝔽3)—23no—
145C2×SL(2,𝔽3)C22×D313yes—
146GL(2,𝔽3)—63no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—43no—
151C3×C22⋊C4—61no—
152C3×C22⋊C4S411yescharacteristic
153C3×C42—31no—
154#217—61no—
155#215—61no—
156#215—61no—
157#215—121no—
158#213—121no—
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—
165C4⋊Dic3—31no—
166C4⋊Dic3—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—
176C2×C3⋊C8—121no—
177C2×C3⋊C8—121no—
178C22×C32—42no—
179C22×C32—41no—
180C22×C32—81no—
181C22×C32—81no—
182C2×C3⋊D3—42no—
183C2×C3⋊D3—81no—
184C2×C3⋊D3—81no—
185C32×C4—81no—
186C32×C4—81no—
187C3⋊Dic3—42no—
188C3⋊Dic3—42no—
189C3⋊Dic3—81no—
190C2×#41C2×C3⋊D311yescharacteristic
191C22×Q8C2×C3⋊D311yescharacteristic
192C2×SD16—91no—
193C2×SD16—91no—
194C2×D8—91no—
195C22×C8—91no—
196C22×C8—91no—
197#126—181no—
198#125—91no—
199#124—31no—
200#123—181no—
201C22⋊Q8—31no—
202#121—31no—
203#121—91no—
204#121—91no—
205#121—181no—
206C4×Q8C2×C3⋊D311yescharacteristic
207C4×D4—31no—
208C4×D4—91no—
209C4×D4—181no—
210#117—91no—
211C2×C4⋊C4—91no—
212#107—91no—
213#106—91no—
214#106—91no—
215C4⋊C8—91no—
216#103—91no—
217#103—91no—
218#103—91no—
219#103—91no—
220#103—91no—
221#103—91no—
222#102—91no—
223#102—91no—
224#102—91no—
225#102—91no—
226C22⋊C8—91no—
227C22⋊C8—91no—
228C4×C8—91no—
229C23×C3—31no—
230C23×C3—43no—
231C23×C3C2×S411yescharacteristic
232C22×D3—61no—
233C22×D3—123no—
234C3×Q8—31no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×Q8#23711yescharacteristic
238C3×D4—61no—
239C3×D4—61no—
240C3×D4—62no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—43no—
245C2×C3×C4—43no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4C2×S411yescharacteristic
251C2×C3×C4C2×S411yescharacteristic
252#67—1212no—
253#67—122no—
254#67—241no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—123no—
259C2×Dic3—123no—
260C2×Dic3—123no—
261C2×Dic3—241no—
262C2×Dic3—246no—
263D12—61no—
264D12—61no—
265C4×D3—241no—
266C4×D3—246no—
267SL(2,𝔽3)#23713yes—
268C3⋊C8—61no—
269C3⋊C8—61no—
270C3⋊C8—61no—
271C3⋊C8—61no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—82no—
276C3⋊D3—82no—
277#41—21no—
278#41—61no—
279C2×Q8—21no—
280C2×Q8—31no—
281C2×Q8—61no—
282C2×Q8C22×C3⋊D311yescharacteristic
283C2×D4—31no—
284C2×D4—31no—
285C2×D4—91no—
286C2×D4—181no—
287C2×D4—181no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—181no—
291C22×C4—181no—
292SD16—181no—
293SD16—181no—
294SD16—361no—
295D8—361no—
296C2×C8—91no—
297C2×C8—91no—
298C2×C8—91no—
299C2×C8—91no—
300C2×C8—361no—
301C2×C8—361no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—181no—
308C4⋊C4—181no—
309C4⋊C4—181no—
310C22⋊C4—31no—
311C22⋊C4—61no—
312C22⋊C4—181no—
313C22⋊C4—181no—
314C22⋊C4—181no—
315C22⋊C4—181no—
316C22⋊C4C3⋊S411yescharacteristic
317C42—31no—
318C42—181no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—46no—
322C22×C3—43no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3—83no—
327C22×C3—83no—
328C22×C3#98512yes—
329C22×C3C22×S411yescharacteristic
330C2×D3—126no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—243no—
335C2×D3—243no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—83no—
343C3×C4—83no—
344Dic3—126no—
345Dic3—126no—
346Dic3—121no—
347Dic3—121no—
348Dic3—121no—
349Dic3—243no—
350C32—41noSylow
351C23—31no—
352C23—181no—
353C23C2×C3⋊S411yescharacteristic
354Q8—31no—
355Q8—61no—
356Q8—61no—
357Q8#383011yescharacteristic
358D4—61no—
359D4—61no—
360D4—62no—
361D4—181no—
362D4—181no—
363D4—362no—
364D4—721no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—721no—
376C2×C4—721no—
377C2×C4C2×C3⋊S411yescharacteristic
378C2×C4C2×C3⋊S411yescharacteristic
379C8—181no—
380C8—181no—
381C8—181no—
382C8—181no—
383C2×C3—22no—
384C2×C3—43no—
385C2×C3—43no—
386C2×C3—43no—
387C2×C3—86no—
388C2×C3—121no—
389C2×C3#591411yescharacteristic
390C2×C3C2×#98511yescharacteristic
391C2×C3#592611yescharacteristic
392D3—246no—
393D3—241no—
394C22—21no—
395C22—21no—
396C22—61no—
397C22—61no—
398C22—61no—
399C22—361no—
400C22—361no—
401C22—361no—
402C22#6426612yes—
403C22C22×C3⋊S411yescharacteristic
404C4—21no—
405C4—21no—
406C4—31no—
407C4—31no—
408C4—61no—
409C4—61no—
410C4—361no—
411C4—361no—
412C4—361no—
413C3—43no—
414C3#8538111yescharacteristic, complemented
415C2—22no—
416C2—121no—
417C2—721no—
418C2#1059657711yescharacteristic
419C2C2×#6426611yescharacteristic
420C2#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. #10593658C2semidirect
4. #10593655C2semidirect
5. C2×C3⋊#859C2semidirect
6. C4×C3⋊GL(2,𝔽3)C2semidirect
7. #10593623C2semidirect
18. C3×#548D3semidirect
20. #548⋊C3D3semidirect
111. #548C3⋊D3semidirect
414. C3#85381semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#105936612
#105936582
#105936552
C2×C3⋊#8592
C4×C3⋊GL(2,𝔽3)2
#105936232
#853813
#835353
#640904