Finite Group Databasebeta

← #11920665

Subgroups — #11920665

Order 1152.

411 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C2×#1007⋊C3C211yesmaximal, characteristic
2#10593654C211yesmaximal, characteristic, complemented
3C3×#5420C211yesmaximal, characteristic, complemented
4#73156—31nomaximal
5#70347—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7#1007⋊C3—21no—
8#1007⋊C3—21no—
9C2×D3×SL(2,𝔽3)C411yescharacteristic, complemented
10C2×C3⋊(#41⋊C3)C411yescharacteristic, complemented
11C3⋊#859—21no—
12C3×#859—21no—
13#63580—41nomaximal
14C2×#1007D311yescharacteristic
15C2×2−1+4⋊C3—31no—
16#5420—31no—
17#5420—61no—
18#5420D311yescharacteristic, complemented
19C3×#419—31no—
20#5225—31no—
21#5139—31no—
22#5132—31no—
23#5006—31no—
24#5005—31no—
25C2×C3×#67—41no—
26C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29D3×SL(2,𝔽3)—21no—
30C3⋊(#41⋊C3)—21no—
31#3757—41no—
32C3×#217—41no—
33#1796—91noSylow
34C22×C3×Q8C2×D311yescharacteristic
35#1007—21no—
36#1007—21no—
37#1007—61no—
38#1007—61no—
39C2×#239—31no—
40C2×#239Dic311yescharacteristic, complemented
41C2×D3×Q8—31no—
42C2×D3×Q8Dic311yescharacteristic, complemented
43C2×#235—31no—
44C2×#235—31no—
452−1+4⋊C3—61no—
462−1+4⋊C3—61no—
47C2×#41⋊C3—31no—
48C22×SL(2,𝔽3)—21no—
49C22×SL(2,𝔽3)—31no—
50C22×SL(2,𝔽3)C2×D311yescharacteristic
51C2×C3×M16—31no—
52C3×#117—31no—
53#926—31no—
54C2×#208—31no—
55#859—21no—
56#859—62no—
57#859—61no—
58C3×#103—61no—
59C3×#103—61no—
60#835—61no—
61#835—61no—
62#834—121no—
63#825—62no—
64#824—61no—
65#823—61no—
66#806—61no—
67#806—61no—
68#805—62no—
69C23×C32—41no—
70C22×C3×D3—41no—
71C2×C3⋊Dic3—81no—
72C3×#67—81no—
73C3×#67—81no—
74C2×C3×Dic3—41no—
75C2×C3×Dic3—81no—
76C3×SL(2,𝔽3)C22⋊C411yescharacteristic
77C2×2−1+4—31no—
78#421—91no—
79#421—91no—
80#419—31no—
81#419—91no—
82#413—91no—
83#410—91no—
84C2×C3×Q8—31no—
85C2×C3×Q8—31no—
86C2×C3×Q8—31no—
87C2×C3×Q8C2×Dic311yescharacteristic
88C2×C3×Q8#6711yescharacteristic
89C2×C3×Q8#6711yescharacteristic
90C2×C3×D4—121no—
91C22×C3×C4—31no—
92C2×#67—31no—
93C2×#67S411yescharacteristic
94#239—21no—
95#239—62no—
96#239—61no—
97#239—61no—
98#239—61no—
99D3×Q8—21no—
100D3×Q8—61no—
101D3×Q8—62no—
102D3×Q8—61no—
103D3×Q8—61no—
104#235—32no—
105#235—32no—
106#235—61no—
107#235—61no—
108#235—61no—
109#235—61no—
110#235—61no—
111#235—61no—
112C2×D12—31no—
113C2×D12—31no—
114C2×C4×D3—31no—
115C2×C4×D3—31no—
116C2×C4×D3—61no—
117C2×Dic6—31no—
118C2×Dic6—31no—
119#41⋊C3—61no—
120C2×SL(2,𝔽3)—21no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—21no—
123C2×SL(2,𝔽3)—61no—
124C2×SL(2,𝔽3)C4×D311yescharacteristic
125C2×SL(2,𝔽3)D1211yescharacteristic
126C2×SL(2,𝔽3)#6711yescharacteristic
127C3×M16—61no—
128C3×M16—61no—
129C2×C3×C8—61no—
130C3×C4⋊C4—61no—
131C3×C22⋊C4—61no—
132C3×C42—61no—
133#217—41no—
134#217—61no—
135#217—121no—
136#217—241no—
137C4⋊Dic3—61no—
138C4×Dic3—61no—
139#208—61no—
140#208—61no—
141C2×C3⋊C8—61no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32—41no—
145C22×C32—161no—
146C2×C3×D3—81no—
147C2×C3×D3—161no—
148C3⋊Dic3—161no—
149C3×Dic3—81no—
150C3×Dic3—161no—
1512−1+4—61no—
1522−1+4—61no—
1532−1+4—181no—
1542−1+4—181no—
155C2×#41—31no—
156C2×#41—91no—
157C2×#41—91no—
158C2×#41—91no—
159C22×Q8—31no—
160C22×Q8—91no—
161C22×Q8D3211yescharacteristic
162C2×M16—31no—
163C2×M16—91no—
164#117—31no—
165#117—91no—
166#104—182no—
167#104—182no—
168#103—61no—
169#103—61no—
170#103—181no—
171#103—181no—
172#101—181no—
173#100—181no—
174#99—181no—
175#99—181no—
176C23×C3—41no—
177C23×C3—81no—
178C23×C3—121no—
179C23×C3C2×S411yescharacteristic
180C22×D3—31no—
181C22×D3SL(2,ℤ/4)11yescharacteristic
182C3×Q8—31no—
183C3×Q8—31no—
184C3×Q8—31no—
185C3×Q8—61no—
186C3×Q8#21711yescharacteristic
187C3×D4—241no—
188C3×D4—241no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—121no—
196C2×C3×C4—121no—
197#67—21no—
198#67—21no—
199#67—61no—
200#67—61no—
201C2×Dic3—31no—
202C2×Dic3—81no—
203C2×Dic3—121no—
204C2×Dic3—241no—
205C2×Dic3—242no—
206C2×Dic3SL(2,ℤ/4)11yescharacteristic
207D12—32no—
208D12—61no—
209D12—61no—
210D12—61no—
211C4×D3—61no—
212C4×D3—61no—
213C4×D3—61no—
214C4×D3—61no—
215C4×D3—62no—
216C4×D3—62no—
217Dic6—32no—
218Dic6—61no—
219Dic6—61no—
220Dic6—61no—
221SL(2,𝔽3)—21no—
222SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
223C3×C8—121no—
224C3⋊C8—121no—
225C2×C32—41no—
226C2×C32—81no—
227C2×C32—81no—
228C2×C32—81no—
229C3×D3—161no—
230#41—61no—
231#41—92no—
232#41—92no—
233#41—181no—
234#41—181no—
235#41—181no—
236#41—181no—
237#41—181no—
238#41—181no—
239#41—182no—
240#41—181no—
241#41—181no—
242#41—181no—
243C2×Q8—31no—
244C2×Q8—31no—
245C2×Q8—31no—
246C2×Q8—61no—
247C2×Q8—91no—
248C2×Q8—91no—
249C2×Q8—181no—
250C2×Q8—181no—
251C2×Q8—182no—
252C2×Q8—181no—
253C2×Q8D3×Dic311yescharacteristic
254C2×Q8#62511yescharacteristic
255C2×Q8#62611yescharacteristic
256C2×D4—31no—
257C2×D4—91no—
258C2×D4—91no—
259C2×D4—91no—
260C22×C4—31no—
261C22×C4—91no—
262C22×C4—91no—
263C22×C4—181no—
264M16—61no—
265M16—61no—
266M16—181no—
267M16—181no—
268C2×C8—61no—
269C2×C8—181no—
270C4⋊C4—61no—
271C4⋊C4—181no—
272C22⋊C4—61no—
273C22⋊C4—181no—
274C42—61no—
275C42—181no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—161no—
284C22×C3—162no—
285C22×C3—241no—
286C22×C3—481no—
287C22×C3C2×SL(2,ℤ/4)11yescharacteristic
288C22×C3GL(2,ℤ/4)11yescharacteristic
289C22×C3GL(2,ℤ/4)11yescharacteristic
290C2×D3—21no—
291C2×D3—41no—
292C2×D3—61no—
293C2×D3—121no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—241no—
299C3×C4—241no—
300Dic3—21no—
301Dic3—61no—
302Dic3—161no—
303Dic3—241no—
304Dic3—481no—
305Dic3—482no—
306C32—41noSylow
307C23—31no—
308C23—91no—
309C23D3×S411yescharacteristic
310Q8—31no—
311Q8—31no—
312Q8—31no—
313Q8—61no—
314Q8—92no—
315Q8—181no—
316Q8—181no—
317Q8—181no—
318Q8(C2×D3)⋊Dic311yescharacteristic
319D4—61no—
320D4—61no—
321D4—92no—
322D4—181no—
323D4—181no—
324D4—181no—
325D4—181no—
326D4—181no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—91no—
335C2×C4—121no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—182no—
341C2×C4—182no—
342C2×C4—361no—
343C8—121no—
344C8—361no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3—81no—
352C2×C3—82no—
353C2×C3—81no—
354C2×C3—161no—
355C2×C3—161no—
356C2×C3—481no—
357C2×C3#543211yescharacteristic
358D3—41no—
359D3—121no—
360C22—41no—
361C22—61no—
362C22—121no—
363C22—181no—
364C22—361no—
365C22D3×SL(2,ℤ/4)11yescharacteristic
366C22#6420911yescharacteristic
367C22#6421011yescharacteristic
368C4—31no—
369C4—31no—
370C4—31no—
371C4—31no—
372C4—61no—
373C4—181no—
374C4—241no—
375C4—721no—
376C3—41no—
377C3—81no—
378C3#7315611yescharacteristic, complemented
379C2—21no—
380C2—21no—
381C2—21no—
382C2—121no—
383C2—361no—
384C2(C2×D3)⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2(C2×D3)⋊SL(2,ℤ/4)
commutatorC2×C3×SL(2,𝔽3)C2×C4
Frattini, hypercenterC23D3×S4
FittingC22×C3×Q8C2×D3
socleC2×C3#5432
prime radicalC2×#1007⋊C3C2
cosocleC22×C3×SL(2,𝔽3)C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #10593654C2semidirect
3. C3×#5420C2semidirect
9. C2×D3×SL(2,𝔽3)C4semidirect
10. C2×C3⋊(#41⋊C3)C4semidirect
18. #5420D3semidirect
40. C2×#239Dic3semidirect
42. C2×D3×Q8Dic3semidirect
378. C3#73156semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
#105936542
C3×#54202
#731563
#703473
#635804