Finite Group Databasebeta

← #11920664

Subgroups — #11920664

Order 1152.

415 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 63 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⋊#226C211yesmaximal, characteristic
2C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×C3×#859C211yesmaximal, characteristic
4#73061—31nomaximal
5#70239—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C3⋊#226—21no—
8C2×C3⋊#226—21no—
9C2×C3⋊#226C412yescomplemented
10Dic3×SL(2,𝔽3)—21no—
11C3×#859—21no—
12#63579—41nomaximal
13C22×#226—31no—
14C22×#226—61no—
15C2×Q8×Dic3D311yescharacteristic
16C22×#216—31no—
17C2×C4×SL(2,𝔽3)—31no—
18C2×#859D311yescharacteristic
19C2×C3×#103—31no—
20C2×#816—31no—
21#4973—31no—
22#4550—31no—
23#4529—31no—
24C22×C3⋊Dic3—41no—
25C22×C3×Dic3—41no—
26C22×C3×Dic3—41no—
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C2×C3×SL(2,𝔽3)D411yescharacteristic
30C3⋊#226—22no—
31#1878—91noSylow
32C22×C3×Q8C2×D311yescharacteristic
33C22×Dic6—31no—
34C22×SL(2,𝔽3)—21no—
35C22×SL(2,𝔽3)C2×D311yescharacteristic
36C2×#226—32no—
37C2×#226—62no—
38C2×#226—62no—
39C2×#226—62no—
40C2×#226—61no—
41C2×#226—61no—
42C22×C3×C8—31no—
43C2×C3×C4⋊C4—31no—
44Q8×Dic3—21no—
45Q8×Dic3—61no—
46C2×#216—32no—
47C2×#216—32no—
48C2×#216—61no—
49C2×#216—61no—
50C2×#216—61no—
51C2×#216—61no—
52C2×C4⋊Dic3—31no—
53C2×C4×Dic3—31no—
54C22×C3⋊C8—31no—
55C4×SL(2,𝔽3)—61no—
56#859—21no—
57C3×#103—61no—
58C3×#103—61no—
59C3⋊#95—61no—
60C3⋊#95—61no—
61C3⋊#95—121no—
62#816—62no—
63C23×C32—41no—
64C2×C3⋊Dic3—42no—
65C2×C3⋊Dic3—81no—
66C2×C3⋊Dic3—81no—
67C2×C3×Dic3—42no—
68C2×C3×Dic3—42no—
69C2×C3×Dic3—81no—
70C2×C3×Dic3—81no—
71C2×C3×Dic3—81no—
72C2×C3×Dic3—81no—
73C3×SL(2,𝔽3)C22⋊C411yescharacteristic
74C22×Q16—91no—
75C2×C4×Q8—31no—
76C2×#103—31no—
77C2×#103—91no—
78#391—91no—
79#340—91no—
80#336—91no—
81C2×C3×Q8—31no—
82C2×C3×Q8—31no—
83C2×C3×Q8—31no—
84C2×C3×Q8C4×D311yescharacteristic
85C2×C3×Q8D1211yescharacteristic
86C2×C3×Q8#6711yescharacteristic
87C22×C3×C4—31no—
88C22×C3×C4—61no—
89C22×C3×C4—121no—
90C22×Dic3—31no—
91C22×Dic3—41no—
92C22×Dic3—61no—
93C22×Dic3—121no—
94C22×Dic3—241no—
95C22×Dic3S411yescharacteristic
96C2×Dic6—32no—
97C2×Dic6—61no—
98C2×Dic6—61no—
99C2×Dic6—61no—
100C2×Dic6—61no—
101C2×Dic6—61no—
102C2×SL(2,𝔽3)—21no—
103C2×SL(2,𝔽3)—21no—
104C2×SL(2,𝔽3)—21no—
105C2×SL(2,𝔽3)C4×D311yescharacteristic
106C2×SL(2,𝔽3)D1211yescharacteristic
107C2×SL(2,𝔽3)#6711yescharacteristic
108#226—64no—
109#226—62no—
110C2×C3×C8—61no—
111C2×C3×C8—61no—
112C2×C3×C8—61no—
113C3×C4⋊C4—61no—
114C3×C4⋊C4—61no—
115#216—62no—
116#216—62no—
117#216—62no—
118#216—62no—
119C4⋊Dic3—32no—
120C4⋊Dic3—61no—
121C4×Dic3—61no—
122C4×Dic3—61no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8—61no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32—41no—
129C22×C32—41no—
130C22×C32—41no—
131C22×C32—41no—
132C22×C32—41no—
133C3⋊Dic3—82no—
134C3×Dic3—82no—
135C3×Dic3—82no—
136C22×Q8—91no—
137C22×Q8D3211yescharacteristic
138C2×Q16—92no—
139C2×Q16—92no—
140C2×Q16—181no—
141C2×Q16—181no—
142C2×Q16—181no—
143C2×Q16—181no—
144C22×C8—31no—
145C22×C8—91no—
146C4×Q8—61no—
147C4×Q8—181no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—91no—
150C2×C42—91no—
151#103—61no—
152#103—61no—
153#103—182no—
154#95—181no—
155#95—181no—
156C23×C3—41no—
157C23×C3—81no—
158C23×C3C2×S411yescharacteristic
159C3×Q8—31no—
160C3×Q8—31no—
161C3×Q8—31no—
162C3×Q8—61no—
163C3×Q8(C2×D3)⋊C411yescharacteristic
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—122no—
171C2×C3×C4—121no—
172C2×C3×C4—121no—
173C2×C3×C4—121no—
174C2×C3×C4—241no—
175C2×C3×C4—241no—
176C2×Dic3—21no—
177C2×Dic3—21no—
178C2×Dic3—42no—
179C2×Dic3—62no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—81no—
184C2×Dic3—81no—
185C2×Dic3—122no—
186C2×Dic3—121no—
187C2×Dic3—121no—
188C2×Dic3—242no—
189C2×Dic3—241no—
190C2×Dic3—241no—
191C2×Dic3—242no—
192C2×Dic3—242no—
193C2×Dic3#22612yescomplemented
194Dic6—62no—
195Dic6—62no—
196Dic6—121no—
197Dic6—121no—
198SL(2,𝔽3)—21no—
199SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
200C3×C8—121no—
201C3⋊C8—62no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×Q8—31no—
210C2×Q8—31no—
211C2×Q8—31no—
212C2×Q8—92no—
213C2×Q8—181no—
214C2×Q8—181no—
215C2×Q8—181no—
216C2×Q8—181no—
217C2×Q8—181no—
218C2×Q8#62411yescharacteristic
219C2×Q8#62611yescharacteristic
220C2×Q8#62611yescharacteristic
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—61no—
224C22×C4—91no—
225C22×C4—181no—
226Q16—182no—
227Q16—182no—
228Q16—182no—
229Q16—182no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—61no—
233C2×C8—92no—
234C2×C8—181no—
235C2×C8—181no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—92no—
239C4⋊C4—181no—
240C42—181no—
241C42—181no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3—81no—
252C22×C3—81no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3C4×S411yescharacteristic
257C22×C3C4⋊S411yescharacteristic
258C22×C3C2×#22611yescharacteristic
259C22×C3#98311yescharacteristic
260C22×C3#98411yescharacteristic
261C22×C3#98511yescharacteristic
262C22×C3GL(2,ℤ/4)11yescharacteristic
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—242no—
268C3×C4—241no—
269Dic3—22no—
270Dic3—82no—
271Dic3—122no—
272Dic3—121no—
273Dic3—242no—
274Dic3—244no—
275C32—41noSylow
276C23D3×S411yescharacteristic
277Q8—31no—
278Q8—31no—
279Q8—31no—
280Q8—61no—
281Q8—182no—
282Q8—182no—
283Q8—361no—
284Q8—361no—
285Q8#372211yescharacteristic
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—32no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—182no—
301C2×C4—181no—
302C2×C4—361no—
303C2×C4—361no—
304C8—121no—
305C8—182no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—81no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3—81no—
317C2×C3—81no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3C4×#22611yescharacteristic
321C2×C3#538811yescharacteristic
322C2×C3C4⋊#22611yescharacteristic
323C2×C3#539511yescharacteristic
324C2×C3(C2×S4)⋊C411yescharacteristic
325C2×C3#541911yescharacteristic
326C2×C3#542511yescharacteristic
327C22#6419611yescharacteristic
328C22#6419711yescharacteristic
329C22D3×#22611yescharacteristic
330C22#6420111yescharacteristic
331C22(C3⋊S4)⋊C411yescharacteristic
332C22#6420711yescharacteristic
333C22#6421011yescharacteristic
334C4—31no—
335C4—31no—
336C4—31no—
337C4—31no—
338C4—62no—
339C4—241no—
340C4—361no—
341C4—362no—
342C3—41no—
343C3—81no—
344C3#7306111yescharacteristic, complemented
345C2#1059319311yescharacteristic
346C2#1059319511yescharacteristic
347C2#1059319711yescharacteristic
348C2#1059320111yescharacteristic
349C2#1059320511yescharacteristic
350C2#1059321111yescharacteristic
351C2#1059324211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D3×S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radical, cosocleC22×C3×SL(2,𝔽3)C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
9. C2×C3⋊#226C4semidirect
193. C2×Dic3#226semidirect
344. C3#73061semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#2262
C2×Dic3×SL(2,𝔽3)2
C2×C3×#8592
#730613
#702393
#635794