Finite Group Databasebeta

← #11922287

Subgroups — #11922287

Order 1152.

551 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 75 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×C3×#985C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4#10593204C212yesmaximal, complemented
5Dic3×#226C212yesmaximal, complemented
6#85365—31nomaximal
7#83898—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#985C414yescomplemented
10C2×C3×GL(2,𝔽3)C2211yescharacteristic
11C2×C3×#226C2211yescharacteristic
12#63841—41nomaximal
13Dic3×SL(2,𝔽3)C2212yes—
14C3⋊#859C2212yes—
15C2×#985D311yescharacteristic
16C2×C3×#135—31no—
17#5827—31no—
18#5827D311yescharacteristic, complemented
19#520⋊C3—31no—
20#5426—31no—
21#5426—61no—
22#5394—32no—
23C4×#226—32no—
24#5232—31no—
25#5232—31no—
26Dic3×Q16—32no—
27#5164—32no—
28#5164—32no—
29Dic3×D8—32no—
30C2×#818—31no—
31#5101—31no—
32C2×C3×C4×D3—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3×GL(2,𝔽3)C2×C412yescomplemented
36C3×#226C2×C412yescomplemented
37C4⋊(C3⋊Dic3)—41no—
38C3×C4⋊Dic3—41no—
39(C2×D3)⋊Dic3—42no—
40Dic32—42no—
41#2951—91noSylow
42C2×C3×#41—31no—
43C2×C3×#41C2×D311yescharacteristic
44C2×#41⋊C3—21no—
45C2×#41⋊C3C2×D311yescharacteristic
46#985Dic314yescomplemented
47C2×GL(2,𝔽3)C2×D311yescharacteristic
48C2×#226C2×D311yescharacteristic
49C3×#135—34no—
50C3×#135—34no—
51C2×C3×Q16—31no—
52C2×C3×SD16—31no—
53C2×C3×SD16—31no—
54C2×C3×D8—31no—
55C22×C3×C8—31no—
56Q8×Dic3—32no—
57Q8×Dic3C2×D312yes—
58D4×Dic3—32no—
59D4×Dic3—32no—
60D4×Dic3—62no—
61#926—31no—
62#926—61no—
63C2×C4⋊Dic3—31no—
64C2×C4⋊Dic3—31no—
65C2×C4⋊Dic3—61no—
66C2×#208—31no—
67#892—121no—
68C4×SL(2,𝔽3)—32no—
69#859—32no—
70#859—62no—
71#835—32no—
72#835—32no—
73#832—32no—
74#832—32no—
75#818—32no—
76#818—32no—
77#815—32no—
78C8×Dic3—32no—
79C22×C3×D3—41no—
80C2×C32×C4—41no—
81C2×C3⋊Dic3—42no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—42no—
84C3×C4×D3—44no—
85C3×SL(2,𝔽3)C22×C411yescharacteristic
86C2×#135—31no—
87#520—31no—
88#520—91no—
89#440—92no—
90#440—92no—
91C4×Q16—92no—
92C4×D8—92no—
93C2×#107—91no—
94#417—91no—
95#417—91no—
96#405—91no—
97C3×#41—32no—
98C3×#41—34no—
99C3×#41—62no—
100C3×#41C4×D312yes—
101C2×C3×Q8—31no—
102C2×C3×Q8C22×D311yescharacteristic
103C2×C3×D4—31no—
104C2×C3×D4—31no—
105C2×C3×D4—61no—
106C22×C3×C4—31no—
107C22×C3×C4—61no—
108C22×Dic3—61no—
109C22×Dic3—62no—
110C22×Dic3—61no—
111C2×C4×D3—41no—
112#41⋊C3—22no—
113#41⋊C3C2×Dic312yescomplemented
114C2×SL(2,𝔽3)—21no—
115C2×SL(2,𝔽3)C22×D311yescharacteristic
116GL(2,𝔽3)C2×Dic312yescomplemented
117#226C2×Dic312yescomplemented
118C3×Q16—32no—
119C3×Q16—32no—
120C3×SD16—34no—
121C3×SD16—32no—
122C3×SD16—32no—
123C3×D8—34no—
124C2×C3×C8—31no—
125C2×C3×C8—31no—
126C2×C3×C8—34no—
127C3×C4⋊C4—121no—
128#217—32no—
129#217—62no—
130(C2×D3)⋊C4—122no—
131C4⋊Dic3—31no—
132C4⋊Dic3—32no—
133C4⋊Dic3—32no—
134C4⋊Dic3—62no—
135C4⋊Dic3—61no—
136C4⋊Dic3—121no—
137C4⋊Dic3—241no—
138C4⋊Dic3S411yescharacteristic
139C4×Dic3—32no—
140C4×Dic3—62no—
141C4×Dic3—122no—
142#208—62no—
143C2×C3⋊C8—32no—
144C22×C32—41no—
145C2×C3×D3—42no—
146C2×C3×D3—81no—
147C2×C3×D3—81no—
148C32×C4—42no—
149C3⋊Dic3—82no—
150C3×Dic3—42no—
151C3×Dic3—82no—
152C2×#41—31no—
153C2×#41D3211yescharacteristic
154#135—34no—
155#135—34no—
156C2×Q16—31no—
157C2×SD16—31no—
158C2×SD16—31no—
159C2×D8—31no—
160C2×M16—91no—
161C22×C8—31no—
162C4×Q8—32no—
163C4×Q8—92no—
164C4×D4—92no—
165C4×D4—92no—
166C4×D4—182no—
167#117—91no—
168#117—181no—
169C2×C4⋊C4—91no—
170C2×C4⋊C4—91no—
171C2×C4⋊C4—181no—
172#107—92no—
173#107—92no—
174#103—92no—
175#103—92no—
176#102—92no—
177#102—92no—
178#97—92no—
179C4×C8—92no—
180C23×C3—31no—
181C23×C3—61no—
182C22×D3—41no—
183C3×Q8—31no—
184C3×Q8—32no—
185C3×Q8—61no—
186C3×Q8C2×C4×D311yescharacteristic
187C3×D4—32no—
188C3×D4—32no—
189C3×D4—61no—
190C3×D4—61no—
191C3×D4—64no—
192C2×C3×C4—31no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—41no—
196C2×C3×C4—61no—
197C2×C3×C4—64no—
198C2×C3×C4—81no—
199C2×C3×C4—122no—
200C2×C3×C4C2×S411yescharacteristic
201C2×Dic3—32no—
202C2×Dic3—41no—
203C2×Dic3—62no—
204C2×Dic3—122no—
205C2×Dic3—242no—
206C2×Dic3—241no—
207C2×Dic3—242no—
208C2×Dic3—241no—
209C2×Dic3C2×S412yes—
210C4×D3—44no—
211SL(2,𝔽3)—21no—
212SL(2,𝔽3)C22×Dic311yescharacteristic
213C3×C8—32no—
214C3×C8—32no—
215C3⋊C8—62no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C3×D3—82no—
220#41—32no—
221#41—34no—
222#41—62no—
223#41D3×Dic312yes—
224C2×Q8—31no—
225C2×Q8C2×D3211yescharacteristic
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—61no—
229C22×C4—31no—
230C22×C4—61no—
231C22×C4—181no—
232C22×C4—182no—
233C22×C4—181no—
234Q16—32no—
235Q16—32no—
236SD16—34no—
237SD16—32no—
238SD16—32no—
239D8—34no—
240M16—182no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—34no—
244C2×C8—92no—
245C4⋊C4—31no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—91no—
249C4⋊C4—182no—
250C4⋊C4—181no—
251C22⋊C4—92no—
252C22⋊C4—182no—
253C42—92no—
254C42—182no—
255C22×C3—32no—
256C22×C3—41no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—62no—
260C22×C3—81no—
261C22×C3—121no—
262C22×C3—121no—
263C22×C3C22×S411yescharacteristic
264C2×D3—42no—
265C2×D3—81no—
266C2×D3—81no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—42no—
270C3×C4—62no—
271C3×C4—82no—
272C3×C4—242no—
273C3×C4C4×S412yes—
274Dic3—22no—
275Dic3—42no—
276Dic3—62no—
277Dic3—122no—
278Dic3—242no—
279Dic3—482no—
280C32—41noSylow
281C23—31no—
282C23—61no—
283Q8—31no—
284Q8—32no—
285Q8—61no—
286Q8C2×D3×Dic311yescharacteristic
287D4—32no—
288D4—32no—
289D4—61no—
290D4—61no—
291D4—64no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—61no—
297C2×C4—64no—
298C2×C4—92no—
299C2×C4—182no—
300C2×C4—721no—
301C2×C4—722no—
302C2×C4—721no—
303C2×C4D3×S411yescharacteristic
304C8—32no—
305C8—32no—
306C8—182no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—62no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3—81no—
314C2×C3—122no—
315C2×C3C2×C4×S411yescharacteristic
316C2×C3#592511yescharacteristic
317C2×C3#592711yescharacteristic
318D3—82no—
319C22—32no—
320C22—61no—
321C22—61no—
322C22—62no—
323C22—121no—
324C22—121no—
325C22C2×D3×S411yescharacteristic
326C4—31no—
327C4—31no—
328C4—62no—
329C4—62no—
330C4—182no—
331C4—362no—
332C4Dic3×S412yes—
333C3—41no—
334C3—81no—
335C3#8536511yescharacteristic, complemented
336C2—62no—
337C2—122no—
338C2#1059644911yescharacteristic
339C2#1059645211yescharacteristic
340C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×C3×#985C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC2×C4D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. #10593660C2semidirect
4. #10593204C2semidirect
5. Dic3×#226C2semidirect
9. C3×#985C4semidirect
18. #5827D3semidirect
33. C3×#41⋊C3C2×C4semidirect
35. C3×GL(2,𝔽3)C2×C4semidirect
36. C3×#226C2×C4semidirect
46. #985Dic3semidirect
113. #41⋊C3C2×Dic3semidirect
116. GL(2,𝔽3)C2×Dic3semidirect
117. #226C2×Dic3semidirect
335. C3#85365semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9852
#5827⋊C32
#105936602
#105932042
Dic3×#2262
#853653
#838983
#638414