Finite Group Databasebeta

← #11923072

Subgroups — #11923072

Order 1152.

834 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#64267C211yesmaximal, characteristic, complemented
3#10593661C212yesmaximal, complemented
4#10593660C211yesmaximal, characteristic, complemented
5#10593655C212yesmaximal, complemented
6#85523—33nomaximal
7#84082—31nomaximal
8C2×C3×#41⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C2×C3×#41⋊C3C2211yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12#64267—22no—
13C2×C3⋊GL(2,𝔽3)C2212yescomplemented
14#64139—41nomaximal
15C3⋊#859C2212yescomplemented
16C22×C3×#41D311yescharacteristic, complemented
17C22×#41⋊C3D313yescomplemented
18C2×#986—33no—
19C2×#949—31no—
20C2×#949—31no—
21C2×#930—31no—
22#5427—36no—
23#5426—33no—
24#5421—36no—
25#5238—32no—
26#5238—32no—
27#5232—31no—
28#5232—31no—
29#5227—32no—
30#5217—32no—
31#5207—31no—
32C22×C32×C4—41no—
33C2×#638—42no—
34C2×C4⋊(C3⋊D3)—41no—
35C3×#41⋊C3—21no—
36C3×#41⋊C3D412yescomplemented
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3⋊GL(2,𝔽3)—22no—
40#3752—42no—
41C4⋊(C3⋊Dic3)—41no—
42#3006—91noSylow
43C2×C3×#41—22no—
44C2×C3×#41—31no—
45C2×C3×#41—31no—
46C2×C3×#41—62no—
47C2×C3×#41C2×D311yescharacteristic
48C2×C3×#41C2×D311yescharacteristic
49C22×C3×Q8C2×D311yescharacteristic
50C22×C3×D4—31no—
51C23×C3×C4—31no—
52C22×#67—61no—
53C22×D12—31no—
54C2×#41⋊C3—26no—
55C2×#41⋊C3C2×D313yes—
56C2×#41⋊C3C2×D313yescomplemented
57C22×SL(2,𝔽3)C2×D313yescomplemented
58#986—66no—
59C2×GL(2,𝔽3)—36no—
60#949—62no—
61#949—64no—
62#949—62no—
63C2×#215—32no—
64C2×#215—32no—
65C2×#213—32no—
66C2×#213—32no—
67#930—32no—
68#930—32no—
69#930—61no—
70#930—61no—
71#930—123no—
72C2×(C2×D3)⋊C4—61no—
73C2×C4⋊Dic3—31no—
74C2×#208—31no—
75C2×#208—31no—
76#859—36no—
77#835—32no—
78#835—32no—
79#832—32no—
80#832—32no—
81#830—32no—
82#830—32no—
83C23×C32—41no—
84C22×C3⋊D3—42no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4—82no—
88#638—84no—
89C2×C3⋊Dic3—42no—
90C4⋊(C3⋊D3)—82no—
91C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
92C22×#41C3⋊D311yescharacteristic, complemented
93C2×AGL(1,ℤ/8)—91no—
94C2×AGL(1,ℤ/8)—91no—
95C2×#121—91no—
96#449—92no—
97#449—92no—
98#448—92no—
99#447—92no—
100#417—91no—
101#417—91no—
102#407—91no—
103C24×C3—31no—
104C23×D3—61no—
105C3×#41—21no—
106C3×#41—32no—
107C3×#41—34no—
108C3×#41—62no—
109C3×#41—62no—
110C3×#41—61no—
111C3×#41—62no—
112C3×#41#6712yescomplemented
113C2×C3×Q8—21no—
114C2×C3×Q8—31no—
115C2×C3×Q8—61no—
116C2×C3×Q8C22×D311yescharacteristic
117C2×C3×D4—31no—
118C2×C3×D4—31no—
119C2×C3×D4—61no—
120C2×C3×D4—62no—
121C2×C3×D4—62no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—43no—
128C22×C3×C4—62no—
129C22×C3×C4—62no—
130C22×C3×C4S411yescharacteristic
131C2×#67—62no—
132C2×#67—62no—
133C2×#67—126no—
134C2×#67—124no—
135C22×Dic3—61no—
136C2×D12—32no—
137C2×D12—62no—
138C2×D12—62no—
139C2×D12—61no—
140C2×D12—123no—
141#41⋊C3—23no—
142#41⋊C3#6716yescomplemented
143C2×SL(2,𝔽3)—23no—
144C2×SL(2,𝔽3)C22×D313yes—
145GL(2,𝔽3)—66no—
146#215—62no—
147#215—64no—
148#215—62no—
149#213—64no—
150#213—64no—
151(C2×D3)⋊C4—62no—
152(C2×D3)⋊C4—62no—
153(C2×D3)⋊C4—126no—
154C4⋊Dic3—32no—
155C4⋊Dic3—61no—
156C4⋊Dic3—123no—
157#208—62no—
158#208—62no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—32no—
161C22×C32—42no—
162C22×C32—41no—
163C22×C32—81no—
164C22×C32—81no—
165C2×C3⋊D3—82no—
166C2×C3⋊D3—82no—
167C2×C3⋊D3—82no—
168C32×C4—42no—
169C32×C4—81no—
170C3⋊Dic3—82no—
171C2×#41—22no—
172C2×#41—31no—
173C2×#41—31no—
174C2×#41—62no—
175C2×#41C2×C3⋊D311yescharacteristic
176C2×#41C2×C3⋊D311yescharacteristic
177C22×Q8C2×C3⋊D311yescharacteristic
178C22×D4—31no—
179C22×D4—91no—
180C22×D4—181no—
181C23×C4—31no—
182AGL(1,ℤ/8)—182no—
183AGL(1,ℤ/8)—184no—
184AGL(1,ℤ/8)—182no—
185C2×SD16—92no—
186C2×SD16—92no—
187C2×D8—92no—
188C2×D8—92no—
189C2×M16—91no—
190C2×M16—91no—
191#121—92no—
192#121—92no—
193#121—181no—
194#121—181no—
195C2×C4⋊C4—91no—
196C2×C22⋊C4—181no—
197#103—92no—
198#103—92no—
199#102—92no—
200#102—92no—
201C22⋊C8—92no—
202C22⋊C8—92no—
203C23×C3—31no—
204C23×C3—31no—
205C23×C3—43no—
206C23×C3—62no—
207C23×C3—62no—
208C23×C3—121no—
209C23×C3C2×S411yescharacteristic
210C22×D3—62no—
211C22×D3—126no—
212C22×D3—122no—
213C22×D3—122no—
214C22×D3—122no—
215C3×Q8—31no—
216C3×Q8—32no—
217C3×Q8—61no—
218C3×Q8C2×#6711yescharacteristic
219C3×D4—32no—
220C3×D4—32no—
221C3×D4—61no—
222C3×D4—61no—
223C3×D4—64no—
224C2×C3×C4—22no—
225C2×C3×C4—31no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—31no—
229C2×C3×C4—43no—
230C2×C3×C4—43no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—62no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—62no—
237C2×C3×C4—86no—
238C2×C3×C4C2×S411yescharacteristic
239C2×C3×C4C2×S411yescharacteristic
240#67—124no—
241#67—124no—
242#67—2412no—
243C2×Dic3—62no—
244C2×Dic3—126no—
245C2×Dic3—122no—
246D12—62no—
247D12—62no—
248D12—122no—
249D12—246no—
250SL(2,𝔽3)C2×#6713yescomplemented
251C3⋊C8—62no—
252C3⋊C8—62no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—82no—
257C3⋊D3—162no—
258C24—31no—
259C24—181no—
260#41—21no—
261#41—32no—
262#41—34no—
263#41—62no—
264#41—62no—
265#41—61no—
266#41—62no—
267#41#63812yescomplemented
268C2×Q8—21no—
269C2×Q8—31no—
270C2×Q8—61no—
271C2×Q8C22×C3⋊D311yescharacteristic
272C2×D4—31no—
273C2×D4—31no—
274C2×D4—61no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—92no—
278C2×D4—182no—
279C2×D4—182no—
280C2×D4—182no—
281C2×D4—182no—
282C2×D4—181no—
283C2×D4—364no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—181no—
292C22×C4C3⋊S411yescharacteristic
293SD16—182no—
294SD16—184no—
295SD16—182no—
296D8—184no—
297D8—184no—
298M16—182no—
299M16—182no—
300C2×C8—92no—
301C2×C8—92no—
302C4⋊C4—92no—
303C4⋊C4—181no—
304C22⋊C4—182no—
305C22⋊C4—182no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—32no—
309C22×C3—46no—
310C22×C3—43no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3—83no—
317C22×C3—83no—
318C22×C3—124no—
319C22×C3#98612yescomplemented
320C22×C3C22×S411yescharacteristic
321C2×D3—122no—
322C2×D3—122no—
323C2×D3—122no—
324C2×D3—246no—
325C2×D3—246no—
326C2×D3—246no—
327C2×D3—244no—
328C3×C4—21no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—46no—
332C3×C4—61no—
333C3×C4—83no—
334C3×C4GL(2,ℤ/4)12yes—
335Dic3—122no—
336Dic3—246no—
337C32—41noSylow
338C23—31no—
339C23—31no—
340C23—62no—
341C23—62no—
342C23—121no—
343C23—182no—
344C23—362no—
345C23—362no—
346C23—362no—
347C23C2×C3⋊S411yescharacteristic
348Q8—31no—
349Q8—32no—
350Q8—61no—
351Q8C2×#63811yescharacteristic
352D4—32no—
353D4—32no—
354D4—61no—
355D4—61no—
356D4—64no—
357D4—182no—
358D4—182no—
359D4—364no—
360D4—364no—
361D4—362no—
362C2×C4—22no—
363C2×C4—31no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—31no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—62no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—62no—
373C2×C4—182no—
374C2×C4—362no—
375C2×C4C2×C3⋊S411yescharacteristic
376C2×C4C2×C3⋊S411yescharacteristic
377C8—182no—
378C8—182no—
379C2×C3—22no—
380C2×C3—43no—
381C2×C3—43no—
382C2×C3—43no—
383C2×C3—62no—
384C2×C3—86no—
385C2×C3—121no—
386C2×C3C2×#98611yescharacteristic
387C2×C3#592311yescharacteristic
388C2×C3C2×GL(2,ℤ/4)11yescharacteristic
389D3—242no—
390D3—486no—
391C22—21no—
392C22—21no—
393C22—32no—
394C22—61no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22—61no—
399C22—124no—
400C22—362no—
401C22—362no—
402C22—362no—
403C22—724no—
404C22#6426712yescomplemented
405C22C22×C3⋊S411yescharacteristic
406C4—21no—
407C4—31no—
408C4—31no—
409C4—61no—
410C4—362no—
411C4#6426912yes—
412C3—43no—
413C3#8552311yescharacteristic, complemented
414C2—22no—
415C2—62no—
416C2—121no—
417C2—722no—
418C2C2×#6426711yescharacteristic
419C2#1059658611yescharacteristic
420C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC22×C3×#41D3
socleC22×C3C22×S4
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
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. C22×C3×#41⋊C3C2semidirect
2. C2×#64267C2semidirect
3. #10593661C2semidirect
4. #10593660C2semidirect
5. #10593655C2semidirect
10. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
13. C2×C3⋊GL(2,𝔽3)C22semidirect
15. C3⋊#859C22semidirect
16. C22×C3×#41D3semidirect
17. C22×#41⋊C3D3semidirect
36. C3×#41⋊C3D4semidirect
56. C2×#41⋊C3C2×D3semidirect
57. C22×SL(2,𝔽3)C2×D3semidirect
91. C3×SL(2,𝔽3)C2×D4semidirect
92. C22×#41C3⋊D3semidirect
112. C3×#41#67semidirect
142. #41⋊C3#67semidirect
250. SL(2,𝔽3)C2×#67semidirect
267. #41#638semidirect
319. C22×C3#986semidirect
404. C22#64267semidirect
413. C3#85523semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#642672
#105936612
#105936602
#105936552
#855233
#840823
#641394