Finite Group Databasebeta

← #11922938

Subgroups — #11922938

Order 1152.

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

normal subgroups only

This group has 576 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#10593660C211yesmaximal, characteristic, complemented
4#10593655C211yesmaximal, characteristic, complemented
5C2×C3⋊#859C211yesmaximal, characteristic, complemented
6#10593628C211yesmaximal, characteristic, complemented
7#10593625C211yesmaximal, characteristic, complemented
8#85389—33nomaximal
9#83536—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
13#64095—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×#548D311yescharacteristic, complemented
19#5555—31no—
20#548⋊C3D313yescomplemented
21#5427—33no—
22#5426—33no—
23#5421—33no—
24C2×#859—33no—
25#5394—33no—
26#5391—33no—
27#5238—31no—
28#5232—31no—
29#5227—31no—
30C2×#835—31no—
31#5057—31no—
32#5055—31no—
33#5049—31no—
34#5044—31no—
35#5038—31no—
36#5033—31no—
37#5026—31no—
38#5021—31no—
39#5011—31no—
40C2×#638—41no—
41C22×C3⋊Dic3—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#3752—41no—
48#3752—41no—
49C4⋊(C3⋊Dic3)—41no—
50C4⋊(C3⋊Dic3)—41no—
51#3308—91noSylow
52C2×C3×#41C2×D311yescharacteristic
53C22×C3×Q8C2×D311yescharacteristic
54C2×#237—61no—
55C2×#41⋊C3C2×D313yes—
56C22×SL(2,𝔽3)C2×D313yes—
57C2×GL(2,𝔽3)—33no—
58C3×#124—31no—
59C3×C22⋊Q8—31no—
60C3×#121—31no—
61C3×C4×Q8C2×D311yescharacteristic
62C3×C4×D4—31no—
63C2×#215—31no—
64C2×#215—31no—
65C2×#213—31no—
66#930—31no—
67#930—31no—
68C2×C4⋊Dic3—31no—
69C2×C4⋊Dic3—31no—
70C2×#208—31no—
71C22×C3⋊C8—31no—
72#897—61no—
73#886—61no—
74#886—61no—
75#886—123no—
76C4×D12—31no—
77#869—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#807—31no—
95#807—31no—
96#807—31no—
97C4⋊(C3⋊C8)—31no—
98#803—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—
107C2×C3⋊Dic3—82no—
108C3×SL(2,𝔽3)#4111yescharacteristic, complemented
109#548C3⋊D311yescharacteristic, complemented
110#547—91no—
111#489—91no—
112#484—91no—
113#480—91no—
114#477—91no—
115#468—91no—
116#466—91no—
117#463—91no—
118#449—91no—
119#448—91no—
120#440—91no—
121#435—91no—
122#417—91no—
123C2×#103—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—
138C22×Dic3—61no—
139C22×Dic3—123no—
140#237—124no—
141C2×D12—31no—
142C2×C4×D3—61no—
143C2×Dic6—61no—
144#41⋊C3—23no—
145C2×SL(2,𝔽3)—23no—
146C2×SL(2,𝔽3)C22×D313yes—
147GL(2,𝔽3)—63no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C22⋊C4—43no—
152C3×C22⋊C4—61no—
153C3×C22⋊C4S411yescharacteristic
154C3×C42—31no—
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—
163(C2×D3)⋊C4—123no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—61no—
168C4⋊Dic3—61no—
169C4⋊Dic3—61no—
170C4⋊Dic3—123no—
171C4⋊Dic3—123no—
172#208—121no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—121no—
178C22×C32—42no—
179C22×C32—41no—
180C22×C32—81no—
181C22×C32—81no—
182C2×C3⋊D3—81no—
183C2×C3⋊D3—81no—
184C2×C3⋊D3—81no—
185C32×C4—81no—
186C32×C4—81no—
187C3⋊Dic3—81no—
188C3⋊Dic3—81no—
189C3⋊Dic3—81no—
190C2×#41—181no—
191C2×#41C2×C3⋊D311yescharacteristic
192C22×Q8C2×C3⋊D311yescharacteristic
193C2×SD16—91no—
194C2×SD16—91no—
195C2×D8—91no—
196C2×M16—91no—
197C22×C8—91no—
198C4⋊Q8—91no—
199#124—31no—
200#123—181no—
201#123—181no—
202C22⋊Q8—31no—
203C22⋊Q8—181no—
204#121—31no—
205#121—91no—
206#121—91no—
207C4×Q8C2×C3⋊D311yescharacteristic
208C4×D4—31no—
209C4×D4—91no—
210C2×C4⋊C4—91no—
211C2×C4⋊C4—91no—
212#107—91no—
213#107—91no—
214#107—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—
228#97—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—122no—
253#67—122no—
254#67—246no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—123no—
259C2×Dic3—123no—
260C2×Dic3—123no—
261C2×Dic3—122no—
262C2×Dic3—122no—
263C2×Dic3—246no—
264D12—61no—
265D12—61no—
266C4×D3—122no—
267Dic6—122no—
268SL(2,𝔽3)#23713yescomplemented
269C3⋊C8—61no—
270C3⋊C8—61no—
271C3⋊C8—61no—
272C3⋊C8—61no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—82no—
277C3⋊D3—161no—
278#41—21no—
279#41—61no—
280#41—364no—
281C2×Q8—21no—
282C2×Q8—31no—
283C2×Q8—61no—
284C2×Q8—181no—
285C2×Q8C22×C3⋊D311yescharacteristic
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—91no—
289C2×D4—181no—
290C2×D4—181no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—181no—
294C22×C4—181no—
295C22×C4—181no—
296SD16—181no—
297SD16—181no—
298SD16—361no—
299D8—361no—
300M16—361no—
301C2×C8—91no—
302C2×C8—91no—
303C2×C8—91no—
304C2×C8—91no—
305C2×C8—361no—
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—91no—
309C4⋊C4—91no—
310C4⋊C4—91no—
311C4⋊C4—181no—
312C4⋊C4—181no—
313C4⋊C4—181no—
314C22⋊C4—31no—
315C22⋊C4—61no—
316C22⋊C4—181no—
317C22⋊C4—181no—
318C22⋊C4—181no—
319C22⋊C4C3⋊S411yescharacteristic
320C42—31no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—46no—
324C22×C3—43no—
325C22×C3—61no—
326C22×C3—61no—
327C22×C3—61no—
328C22×C3—83no—
329C22×C3—83no—
330C22×C3#98612yes—
331C22×C3C22×S411yescharacteristic
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—243no—
336C2×D3—243no—
337C2×D3—243no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—83no—
345C3×C4—83no—
346Dic3—121no—
347Dic3—121no—
348Dic3—121no—
349Dic3—243no—
350Dic3—243no—
351Dic3—243no—
352C32—41noSylow
353C23—31no—
354C23—181no—
355C23C2×C3⋊S411yescharacteristic
356Q8—31no—
357Q8—61no—
358Q8—61no—
359Q8—362no—
360Q8#383011yescharacteristic
361D4—61no—
362D4—61no—
363D4—62no—
364D4—181no—
365D4—181no—
366D4—362no—
367D4—362no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4—362no—
379C2×C4—362no—
380C2×C4—362no—
381C2×C4C2×C3⋊S411yescharacteristic
382C2×C4C2×C3⋊S411yescharacteristic
383C8—181no—
384C8—181no—
385C8—181no—
386C8—181no—
387C2×C3—22no—
388C2×C3—43no—
389C2×C3—43no—
390C2×C3—43no—
391C2×C3—86no—
392C2×C3—121no—
393C2×C3#591411yescharacteristic
394C2×C3C2×#98611yescharacteristic
395C2×C3#592711yescharacteristic
396D3—241no—
397D3—483no—
398C22—21no—
399C22—21no—
400C22—61no—
401C22—61no—
402C22—61no—
403C22—361no—
404C22—361no—
405C22—361no—
406C22#6426712yes—
407C22C22×C3⋊S411yescharacteristic
408C4—21no—
409C4—21no—
410C4—31no—
411C4—31no—
412C4—61no—
413C4—61no—
414C4—361no—
415C4—361no—
416C4—361no—
417C3—43no—
418C3#8538911yescharacteristic, complemented
419C2—22no—
420C2—121no—
421C2—721no—
422C2#1059657711yescharacteristic
423C2C2×#6426711yescharacteristic
424C2#1059659011yescharacteristic
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. C2×C3⋊#859C2semidirect
6. #10593628C2semidirect
7. #10593625C2semidirect
18. C3×#548D3semidirect
20. #548⋊C3D3semidirect
108. C3×SL(2,𝔽3)#41semidirect
109. #548C3⋊D3semidirect
268. SL(2,𝔽3)#237semidirect
418. C3#85389semidirect

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
C2×C3⋊#8592
#105936282
#105936252
#853893
#835363
#640954