Finite Group Databasebeta

← #11921113

Subgroups — #11921113

Order 1152.

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

normal subgroups only

This group has 458 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×#5945C211yesmaximal, characteristic
2#10593658C211yesmaximal, characteristic, complemented
3#10593657C211yesmaximal, characteristic, complemented
4#73157—33nomaximal
5#72146—31nomaximal
6C3×2−1+4⋊C3—21no—
7C3×2−1+4⋊C3C411yescharacteristic, complemented
8C3×#994C411yescharacteristic, complemented
9C2×C3×#41⋊C3—21no—
10C2×C3×#41⋊C3C2211yescharacteristic
11C3⋊GU(2,𝔽3)—21no—
12C3⋊#859—21no—
13#63664—41nomaximal
14C3×#585D311yescharacteristic
15#5945D313yes—
16#5424—33no—
17#5423—33no—
18#5240—31no—
19#5236—31no—
20#5236—31no—
21#5234—31no—
22#5233—31no—
23#5233—31no—
24C32×#41—41no—
25C3×#41⋊C3—21no—
26C3×#41⋊C3C2×C411yescharacteristic
27C3×#41⋊C3D411yescharacteristic
28C2×C3×SL(2,𝔽3)—21no—
29C2×C3×SL(2,𝔽3)D411yescharacteristic
30C4×C3⋊Dic3—41no—
31#3748—41no—
32#1794—91noSylow
33C3×2−1+4—21no—
34C3×2−1+4—31no—
35C3×2−1+4Dic311yescharacteristic, complemented
36C3×H(5,𝔽2)—31no—
37C3×H(5,𝔽2)—61no—
38C3×H(5,𝔽2)Dic311yescharacteristic, complemented
39C2×C3×#41—21no—
40C2×C3×#41—31no—
41C2×C3×#41—31no—
42C2×C3×#41—61no—
43C2×C3×#41C2×D311yescharacteristic
442−1+4⋊C3—23no—
452−1+4⋊C3Dic313yescomplemented
46#994Dic313yescomplemented
47C2×#41⋊C3—23no—
48C2×#41⋊C3C2×D313yes—
49#926—31no—
50#926—31no—
51C2×#208—31no—
52C22×C3⋊C8—31no—
53GU(2,𝔽3)—63no—
54#859—63no—
55#837—61no—
56#837—61no—
57#837—61no—
58#837—61no—
59#837—123no—
60#835—61no—
61#835—61no—
62#834—62no—
63#832—61no—
64#832—61no—
65#830—62no—
66C32×Q8—41no—
67C32×D4—41no—
68C32×D4—81no—
69C2×C32×C4—41no—
70C2×C32×C4—81no—
71C2×C3⋊Dic3—81no—
72C3×SL(2,𝔽3)C22⋊C411yescharacteristic
73C3⋊(C3⋊C8)—81no—
74#585C3⋊D311yescharacteristic
75#421—91no—
76#421—91no—
77#416—91no—
78#416—91no—
79#410—91no—
80#408—91no—
81C3×#41—21no—
82C3×#41—31no—
83C3×#41—31no—
84C3×#41—31no—
85C3×#41—31no—
86C3×#41—31no—
87C3×#41—43no—
88C3×#41—61no—
89C3×#41—61no—
90C3×#41—61no—
91C3×#41—61no—
92C3×#41—62no—
93C3×#41—61no—
94C3×#41—61no—
95C3×#41—61no—
96C3×#41—61no—
97C3×#41C2×Dic311yescharacteristic
98C3×#41#6711yescharacteristic
99C3×#41S411yescharacteristic
100C2×C3×Q8—21no—
101C2×C3×Q8—31no—
102C2×C3×Q8—31no—
103C2×C3×Q8—61no—
104C2×C3×Q8#6711yescharacteristic
105C2×C3×D4—31no—
106C2×C3×D4—31no—
107C2×C3×D4—31no—
108C2×C3×D4—32no—
109C2×C3×D4—61no—
110C2×C3×D4—61no—
111C2×C3×D4—61no—
112C2×C3×D4—62no—
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C22×C3×C4—61no—
116#41⋊C3—23no—
117#41⋊C3C2×Dic313yes—
118#41⋊C3#6713yes—
119C2×SL(2,𝔽3)—23no—
120C2×SL(2,𝔽3)#6713yes—
121#217—61no—
122#217—61no—
123C4⋊Dic3—61no—
124C4⋊Dic3—61no—
125C4×Dic3—61no—
126C4×Dic3—61no—
127C4×Dic3—123no—
128#208—61no—
129#208—61no—
130#208—123no—
131C2×C3⋊C8—61no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—61no—
134C2×C3⋊C8—61no—
135C22×C32—41no—
136C22×C32—81no—
137C32×C4—41no—
138C32×C4—41no—
139C32×C4—81no—
140C3⋊Dic3—82no—
1412−1+4—21no—
1422−1+4—31no—
1432−1+4C3⋊Dic311yescharacteristic, complemented
144H(5,𝔽2)—31no—
145H(5,𝔽2)—61no—
146H(5,𝔽2)C3⋊Dic311yescharacteristic, complemented
147C2×#41—21no—
148C2×#41—31no—
149C2×#41—31no—
150C2×#41—61no—
151C2×#41C2×C3⋊D311yescharacteristic
152C2×M16—91no—
153C22×C8—91no—
154#117—91no—
155#117—91no—
156#104—181no—
157#104—181no—
158#104—181no—
159#104—181no—
160#103—181no—
161#103—181no—
162#102—181no—
163#102—181no—
164#99—182no—
165C22⋊C8—182no—
166C23×C3—31no—
167C23×C3—32no—
168C23×C3—61no—
169C3×Q8—31no—
170C3×Q8—31no—
171C3×Q8—43no—
172C3×Q8—61no—
173C3×Q8—61no—
174C3×Q8#21711yescharacteristic
175C3×Q8SL(2,ℤ/4)11yescharacteristic
176C3×D4—21no—
177C3×D4—31no—
178C3×D4—31no—
179C3×D4—31no—
180C3×D4—43no—
181C3×D4—61no—
182C3×D4—61no—
183C3×D4—61no—
184C3×D4—61no—
185C3×D4—61no—
186C3×D4—62no—
187C3×D4—61no—
188C3×D4—83no—
189C3×D4SL(2,ℤ/4)11yescharacteristic
190C2×C3×C4—21no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—43no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—62no—
202C2×C3×C4—61no—
203C2×C3×C4—83no—
204C2×C3×C4C2×S411yescharacteristic
205C2×Dic3—121no—
206C2×Dic3—121no—
207C2×Dic3—243no—
208SL(2,𝔽3)#21713yes—
209C3⋊C8—121no—
210C3⋊C8—121no—
211C3⋊C8—243no—
212C2×C32—41no—
213C2×C32—81no—
214C2×C32—161no—
215#41—21no—
216#41—31no—
217#41—31no—
218#41—31no—
219#41—31no—
220#41—31no—
221#41—61no—
222#41—61no—
223#41—61no—
224#41—61no—
225#41—62no—
226#41—61no—
227#41—61no—
228#41—61no—
229#41—61no—
230#41C2×C3⋊Dic311yescharacteristic
231#41#63811yescharacteristic
232#41C3⋊S411yescharacteristic
233C2×Q8—21no—
234C2×Q8—31no—
235C2×Q8—31no—
236C2×Q8—61no—
237C2×Q8#63811yescharacteristic
238C2×D4—31no—
239C2×D4—31no—
240C2×D4—31no—
241C2×D4—32no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—62no—
246C22×C4—31no—
247C22×C4—32no—
248C22×C4—61no—
249M16—181no—
250M16—181no—
251C2×C8—181no—
252C2×C8—181no—
253C2×C8—181no—
254C2×C8—181no—
255C4⋊C4—181no—
256C4⋊C4—181no—
257C22⋊C4—181no—
258C22⋊C4—181no—
259C42—181no—
260C42—181no—
261C22×C3—21no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—43no—
265C22×C3—61no—
266C22×C3—83no—
267C22×C3—121no—
268C22×C3—122no—
269C22×C3—241no—
270C22×C3GL(2,ℤ/4)11yescharacteristic
271C3×C4—21no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—43no—
275C3×C4—43no—
276C3×C4—61no—
277C3×C4—83no—
278C3×C4C2×SL(2,ℤ/4)11yescharacteristic
279C3×C4GL(2,ℤ/4)11yescharacteristic
280Dic3—241no—
281Dic3—241no—
282Dic3—246no—
283C32—41noSylow
284C23—31no—
285C23—32no—
286C23—61no—
287Q8—31no—
288Q8—31no—
289Q8—61no—
290Q8—61no—
291Q8#375711yescharacteristic
292Q8C3⋊SL(2,ℤ/4)11yescharacteristic
293D4—21no—
294D4—31no—
295D4—31no—
296D4—31no—
297D4—61no—
298D4—61no—
299D4—61no—
300D4—61no—
301D4—61no—
302D4—62no—
303D4—61no—
304D4C3⋊SL(2,ℤ/4)11yescharacteristic
305C2×C4—21no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—32no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—62no—
316C2×C4—61no—
317C2×C4—361no—
318C2×C4—361no—
319C2×C4C2×C3⋊S411yescharacteristic
320C8—361no—
321C8—361no—
322C2×C3—21no—
323C2×C3—41no—
324C2×C3—43no—
325C2×C3—61no—
326C2×C3—61no—
327C2×C3—83no—
328C2×C3—121no—
329C2×C3—163no—
330C2×C3#543211yescharacteristic
331C22—21no—
332C22—31no—
333C22—31no—
334C22—61no—
335C22—121no—
336C22—122no—
337C22—241no—
338C22#6426911yescharacteristic
339C4—21no—
340C4—31no—
341C4—31no—
342C4—61no—
343C4—721no—
344C4—721no—
345C4C2×C3⋊SL(2,ℤ/4)11yescharacteristic
346C4#6426911yescharacteristic
347C3—43no—
348C3#7315711yescharacteristic, complemented
349C2—21no—
350C2—41no—
351C2—61no—
352C2—61no—
353C2—121no—
354C2#1059366611yescharacteristic
subgroupG/H
centerC4#64269
commutatorC3×#41⋊C3C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC3×#585D3
socleC2×C3#5432
prime radicalC3×#5945C2
cosocleC2×C3×#41⋊C3C22
hypercenter#41C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #10593658C2semidirect
3. #10593657C2semidirect
7. C3×2−1+4⋊C3C4semidirect
8. C3×#994C4semidirect
35. C3×2−1+4Dic3semidirect
38. C3×H(5,𝔽2)Dic3semidirect
45. 2−1+4⋊C3Dic3semidirect
46. #994Dic3semidirect
143. 2−1+4C3⋊Dic3semidirect
146. H(5,𝔽2)C3⋊Dic3semidirect
348. C3#73157semidirect

Maximal subgroups

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

subgroupindex
C3×#59452
#105936582
#105936572
#731573
#721463
#636644