Finite Group Databasebeta

← #11922943

Subgroups — #11922943

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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593655C211yesmaximal, characteristic, complemented
3#10593655C211yesmaximal, characteristic, complemented
4#10593654C211yesmaximal, characteristic, complemented
5C2×C3⋊#859C211yesmaximal, characteristic, complemented
6#10593628C211yesmaximal, characteristic, complemented
7#10593623C211yesmaximal, characteristic, complemented
8#85401—33nomaximal
9#83553—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
13#64148—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×D4×Q8D311yescharacteristic, complemented
19#5565—31no—
20D4×SL(2,𝔽3)D313yescomplemented
21#5421—33no—
22#5421—33no—
23#5420—33no—
24C2×#859—33no—
25#5394—33no—
26#5389—33no—
27#5227—31no—
28#5227—31no—
29#5225—31no—
30C2×#835—31no—
31#5083—31no—
32#5082—31no—
33#5049—31no—
34#5048—31no—
35#5045—31no—
36#5044—31no—
37#5026—31no—
38#5022—31no—
39#5011—31no—
40C2×C32×D4—41no—
41C2×#638—41no—
42C2×#638—41no—
43C22×C3⋊Dic3—41no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)C2311yescharacteristic
47C3⋊GL(2,𝔽3)—21no—
48#3757—41no—
49#3752—41no—
50#3750—41no—
51#3320—91noSylow
52C22×C3×Q8C2×D311yescharacteristic
53C22×C3×Q8C2×D311yescharacteristic
54C22×SL(2,𝔽3)C2×D313yes—
55C22×SL(2,𝔽3)C2×D313yes—
56C2×GL(2,𝔽3)—33no—
57C3×C4⋊Q8—31no—
58C3×C22⋊Q8—31no—
59C3×C22⋊Q8—31no—
60C3×C4×Q8C2×D311yescharacteristic
61C3×C4×D4—31no—
62C2×#215—31no—
63C2×#215—31no—
64C2×#215—31no—
65#939—61no—
66#939—123no—
67#930—31no—
68#930—31no—
69#926—31no—
70C2×C4⋊Dic3—31no—
71C2×#208—31no—
72C22×C3⋊C8—31no—
73#898—61no—
74#883—61no—
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#835—31no—
89#835—31no—
90#835—31no—
91#830—31no—
92#830—31no—
93#809—31no—
94#808—31no—
95#807—31no—
96#807—31no—
97C4⋊(C3⋊C8)—31no—
98#803—31no—
99C23×C32—41no—
100C23×C32—41no—
101C22×C3⋊D3—41no—
102C32×D4—82no—
103C2×C32×C4—41no—
104#638—82no—
105#638—82no—
106C2×C3⋊Dic3—41no—
107C2×C3⋊Dic3—41no—
108C2×C3⋊Dic3—41no—
109C2×C3⋊Dic3—82no—
110C3×SL(2,𝔽3)#4111yescharacteristic
111#553—91no—
112D4×Q8C3⋊D311yescharacteristic, complemented
113#491—91no—
114#485—91no—
115#484—91no—
116#479—91no—
117#468—91no—
118#465—91no—
119#461—91no—
120#448—91no—
121#448—91no—
122#440—91no—
123#435—91no—
124#419—91no—
125C2×#103—91no—
126C2×C3×Q8—21no—
127C2×C3×Q8—21no—
128C2×C3×Q8—31no—
129C2×C3×Q8—31no—
130C2×C3×Q8—61no—
131C2×C3×Q8—61no—
132C2×C3×Q8C22×D311yescharacteristic
133C2×C3×D4—43no—
134C2×C3×D4S411yescharacteristic
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C2×#67—61no—
138C2×#67—61no—
139C2×#67—123no—
140C2×#67—123no—
141C22×Dic3—61no—
142C22×Dic3—123no—
143C2×D12—31no—
144C2×C4×D3—61no—
145C2×SL(2,𝔽3)—23no—
146C2×SL(2,𝔽3)—23no—
147C2×SL(2,𝔽3)C22×D313yes—
148GL(2,𝔽3)—63no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—31no—
151C3×C4⋊C4—61no—
152C3×C22⋊C4—31no—
153C3×C22⋊C4—31no—
154C3×C42—31no—
155#217—61no—
156#217—123no—
157#215—61no—
158#215—61no—
159#215—121no—
160#215—121no—
161(C2×D3)⋊C4—61no—
162(C2×D3)⋊C4—61no—
163(C2×D3)⋊C4—61no—
164(C2×D3)⋊C4—123no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—61no—
169Dic3⋊C4—61no—
170Dic3⋊C4—61no—
171Dic3⋊C4—123no—
172C4×Dic3—61no—
173#208—121no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—121no—
179C22×C32—42no—
180C22×C32—41no—
181C22×C32—81no—
182C22×C32—81no—
183C22×C32—81no—
184C22×C32—81no—
185C22×C32—81no—
186C2×C3⋊D3—81no—
187C2×C3⋊D3—81no—
188C2×C3⋊D3—81no—
189C32×C4—81no—
190C3⋊Dic3—42no—
191C3⋊Dic3—81no—
192C3⋊Dic3—81no—
193C22×Q8C2×C3⋊D311yescharacteristic
194C22×Q8C2×C3⋊D311yescharacteristic
195C2×SD16—91no—
196C2×SD16—91no—
197C2×SD16—91no—
198C2×M16—91no—
199C22×C8—91no—
200C4⋊Q8—31no—
201#126—181no—
202#125—91no—
203#123—181no—
204C22⋊Q8—31no—
205C22⋊Q8—31no—
206#121—91no—
207#121—91no—
208#121—181no—
209C4×Q8C2×C3⋊D311yescharacteristic
210C4×D4—31no—
211C4×D4—91no—
212C4×D4—181no—
213#117—91no—
214C2×C4⋊C4—91no—
215#107—91no—
216#107—91no—
217#106—91no—
218C4⋊C8—91no—
219#103—91no—
220#103—91no—
221#103—91no—
222#103—91no—
223#103—91no—
224#103—91no—
225#103—91no—
226#103—91no—
227#103—91no—
228#102—91no—
229C22⋊C8—91no—
230C22⋊C8—91no—
231#97—91no—
232C23×C3—43no—
233C23×C3—43no—
234C23×C3C2×S411yescharacteristic
235C23×C3C2×S411yescharacteristic
236C22×D3—61no—
237C22×D3—123no—
238C3×Q8—31no—
239C3×Q8—61no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×Q8—61no—
243C3×Q8#23711yescharacteristic
244C3×D4—22no—
245C3×D4—86no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—43no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4C2×S411yescharacteristic
256#67—122no—
257#67—246no—
258#67—241no—
259#67—246no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—123no—
264C2×Dic3—123no—
265C2×Dic3—123no—
266C2×Dic3—241no—
267C2×Dic3—246no—
268D12—61no—
269D12—61no—
270C4×D3—241no—
271SL(2,𝔽3)#23713yes—
272C3⋊C8—61no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—61no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—82no—
280C2×C32—161no—
281C3⋊D3—161no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—31no—
285C2×Q8—31no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8C22×C3⋊D311yescharacteristic
289C2×D4—91no—
290C2×D4—181no—
291C2×D4—181no—
292C2×D4C3⋊S411yescharacteristic
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—181no—
296C22×C4—181no—
297SD16—181no—
298SD16—181no—
299SD16—361no—
300SD16—361no—
301M16—361no—
302C2×C8—91no—
303C2×C8—91no—
304C2×C8—91no—
305C2×C8—91no—
306C2×C8—361no—
307C4⋊C4—31no—
308C4⋊C4—31no—
309C4⋊C4—61no—
310C4⋊C4—91no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C4⋊C4—181no—
314C4⋊C4—181no—
315C4⋊C4—181no—
316C22⋊C4—31no—
317C22⋊C4—31no—
318C22⋊C4—181no—
319C22⋊C4—181no—
320C22⋊C4—181no—
321C22⋊C4—181no—
322C42—31no—
323C42—181no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—46no—
330C22×C3—43no—
331C22×C3—83no—
332C22×C3—83no—
333C22×C3—83no—
334C22×C3—83no—
335C22×C3—83no—
336C22×C3#98312yes—
337C22×C3C22×S411yescharacteristic
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—243no—
342C2×D3—243no—
343C2×D3—243no—
344C3×C4—21no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—83no—
351Dic3—126no—
352Dic3—121no—
353Dic3—121no—
354Dic3—121no—
355Dic3—243no—
356Dic3—243no—
357C32—41noSylow
358C23—181no—
359C23C2×C3⋊S411yescharacteristic
360C23C2×C3⋊S411yescharacteristic
361Q8—31no—
362Q8—61no—
363Q8—61no—
364Q8—61no—
365Q8—61no—
366Q8#383011yescharacteristic
367D4—22no—
368D4—181no—
369D4—181no—
370D4—362no—
371D4—721no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—721no—
384C2×C4—721no—
385C2×C4C2×C3⋊S411yescharacteristic
386C8—181no—
387C8—181no—
388C8—181no—
389C8—181no—
390C2×C3—22no—
391C2×C3—41no—
392C2×C3—43no—
393C2×C3—43no—
394C2×C3—43no—
395C2×C3—86no—
396C2×C3—163no—
397C2×C3#591411yescharacteristic
398C2×C3C2×#98311yescharacteristic
399C2×C3#592611yescharacteristic
400D3—241no—
401D3—483no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—361no—
408C22—361no—
409C22—361no—
410C22#6426412yes—
411C22C22×C3⋊S411yescharacteristic
412C4—21no—
413C4—31no—
414C4—31no—
415C4—61no—
416C4—61no—
417C4—61no—
418C4—361no—
419C4—361no—
420C4—361no—
421C3—43no—
422C3#8540111yescharacteristic, complemented
423C2—22no—
424C2—41no—
425C2—721no—
426C2#1059657711yescharacteristic
427C2C2×#6426411yescharacteristic
428C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×D4×Q8D3
socleC22×C3C22×S4
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
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. C3×D4×SL(2,𝔽3)C2semidirect
2. #10593655C2semidirect
3. #10593655C2semidirect
4. #10593654C2semidirect
5. C2×C3⋊#859C2semidirect
6. #10593628C2semidirect
7. #10593623C2semidirect
18. C3×D4×Q8D3semidirect
20. D4×SL(2,𝔽3)D3semidirect
112. D4×Q8C3⋊D3semidirect
422. C3#85401semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
#105936552
#105936542
C2×C3⋊#8592
#105936282
#105936232
#854013
#835533
#641484