Finite Group Databasebeta

← #11923772

Subgroups — #11923772

Order 1152.

600 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 35 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊#5931C211yesmaximal, characteristic, complemented
3C3⋊#5930C211yesmaximal, characteristic, complemented
4#87507—33nomaximal
5#84059—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7#64269—41nomaximal
8#64264—42nomaximal
9C23×C3×Q8D311yescharacteristic, complemented
10C2×C22⋊SL(2,𝔽3)D313yescomplemented
11#5931—33no—
12#5930—33no—
13C2×#942—32no—
14#5791—31no—
15#5228—31no—
16#5228—32no—
17#5228—31no—
18#5227—31no—
19#5227—32no—
20#5227—31no—
21#5225—32no—
22#5207—31no—
23C22×C3×A4—41no—
24C2×C3⋊S4—41no—
25C2×C3×SL(2,𝔽3)—42no—
26C2×C3×SL(2,𝔽3)—81no—
27C3⋊SL(2,ℤ/4)—41no—
28C3⋊GL(2,𝔽3)—42no—
29C3⋊#226—42no—
30#3004—91noSylow
31C22×C3×Q8—31no—
32C22×C3×Q8—32no—
33C22×C3×Q8—32no—
34C22×C3×Q8—61no—
35C22×C3×Q8—61no—
36C22×C3×Q8C2×D311yescharacteristic
37C23×C3×C4—31no—
38C2×#235—31no—
39C22⋊SL(2,𝔽3)C2×D313yes—
40GL(2,ℤ/4)—123no—
41#983—126no—
42#953—61no—
43C2×#216—32no—
44C2×#216—32no—
45#942—62no—
46#942—62no—
47#942—62no—
48#942—62no—
49C2×#215—32no—
50C2×#215—32no—
51#930—31no—
52#930—31no—
53#929—61no—
54C4×#67—61no—
55C4×#67—61no—
56#926—31no—
57#924—31no—
58#924—31no—
59C2×#208—32no—
60#835—32no—
61#835—32no—
62#835—32no—
63#835—32no—
64#830—32no—
65#830—31no—
66#830—31no—
67C2×C3×A4—41no—
68C2×C3×A4—81no—
69C3⋊S4—81no—
70#638—161no—
71C3×SL(2,𝔽3)—42no—
72C3×SL(2,𝔽3)—81no—
73C23×Q8C3⋊D311yescharacteristic, complemented
74C2×#137—92no—
75#525—91no—
76#451—91no—
77#451—92no—
78#451—91no—
79#448—91no—
80#448—92no—
81#448—91no—
82#419—92no—
83#407—91no—
84C24×C3S411yescharacteristic
85C22×A4—43no—
86C2×S4—123no—
87C2×C3×Q8—21no—
88C2×C3×Q8—32no—
89C2×C3×Q8—32no—
90C2×C3×Q8—32no—
91C2×C3×Q8—32no—
92C2×C3×Q8—32no—
93C2×C3×Q8—61no—
94C2×C3×Q8—61no—
95C2×C3×Q8—62no—
96C2×C3×Q8—62no—
97C2×C3×Q8—62no—
98C2×C3×Q8—61no—
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×Q8—62no—
102C2×C3×Q8S412yescomplemented
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—62no—
110C22×C3×C4—62no—
111C2×#67—61no—
112C2×#67—61no—
113C2×#67—61no—
114#235—62no—
115#235—62no—
116C2×D12—31no—
117C2×C4×D3—61no—
118C2×Dic6—31no—
119C2×SL(2,𝔽3)—46no—
120C2×SL(2,𝔽3)—83no—
121SL(2,ℤ/4)—123no—
122GL(2,𝔽3)—126no—
123#226—126no—
124#217—61no—
125#217—61no—
126#217—61no—
127#216—62no—
128#216—62no—
129#216—62no—
130#216—62no—
131#215—62no—
132#215—62no—
133#215—62no—
134#215—62no—
135(C2×D3)⋊C4—61no—
136(C2×D3)⋊C4—61no—
137C4⋊Dic3—31no—
138C4⋊Dic3—31no—
139Dic3⋊C4—61no—
140Dic3⋊C4—61no—
141C4×Dic3—61no—
142#208—62no—
143#208—62no—
144C2×C3⋊C8—32no—
145C2×C3⋊C8—32no—
146C22×C32—161no—
147C2×C3⋊D3—161no—
148C3×A4—41no—
149C3⋊Dic3—161no—
150C2×#41—91no—
151C22×Q8—31no—
152C22×Q8—32no—
153C22×Q8—32no—
154C22×Q8—61no—
155C22×Q8—61no—
156C22×Q8C2×C3⋊D311yescharacteristic
157C23×C4—31no—
158#137—182no—
159#137—182no—
160#137—182no—
161#137—182no—
162C2×Q16—92no—
163C2×Q16—92no—
164C2×SD16—92no—
165C2×SD16—92no—
166C2×M16—92no—
167#123—181no—
168C22⋊Q8—91no—
169C22⋊Q8—91no—
170#121—91no—
171#121—91no—
172#120—181no—
173C4×D4—181no—
174C4×D4—181no—
175#117—91no—
176#103—92no—
177#103—92no—
178#103—92no—
179#103—92no—
180C22⋊C8—92no—
181C22⋊C8—91no—
182C22⋊C8—91no—
183C23×C3—21no—
184C23×C3—31no—
185C23×C3—31no—
186C23×C3—61no—
187C23×C3C2×S411yescharacteristic
188C22×D3—61no—
189C2×A4—43no—
190C2×A4—83no—
191S4—243no—
192C3×Q8—21no—
193C3×Q8—32no—
194C3×Q8—32no—
195C3×Q8—32no—
196C3×Q8—61no—
197C3×Q8—62no—
198C3×Q8—61no—
199C3×Q8—62no—
200C3×Q8—61no—
201C3×Q8C2×S412yes—
202C2×C3×C4—31no—
203C2×C3×C4—32no—
204C2×C3×C4—31no—
205C2×C3×C4—32no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214#67—121no—
215#67—121no—
216#67—122no—
217#67—121no—
218#67—121no—
219#67—483no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223D12—62no—
224C4×D3—122no—
225Dic6—62no—
226SL(2,𝔽3)—46no—
227SL(2,𝔽3)—83no—
228C3⋊C8—62no—
229C3⋊C8—62no—
230C2×C32—161no—
231C2×C32—321no—
232C3⋊D3—321no—
233C24C3⋊S411yescharacteristic
234#41—182no—
235#41—182no—
236C2×Q8—21no—
237C2×Q8—32no—
238C2×Q8—32no—
239C2×Q8—32no—
240C2×Q8—32no—
241C2×Q8—32no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—62no—
245C2×Q8—62no—
246C2×Q8—62no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—62no—
251C2×Q8—91no—
252C2×Q8C3⋊S412yescomplemented
253C2×D4—91no—
254C2×D4—181no—
255C2×D4—181no—
256C2×D4—181no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—181no—
266Q16—182no—
267Q16—182no—
268Q16—182no—
269Q16—182no—
270SD16—182no—
271SD16—182no—
272SD16—182no—
273SD16—182no—
274M16—182no—
275M16—182no—
276C2×C8—92no—
277C2×C8—92no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—181no—
281C4⋊C4—181no—
282C22⋊C4—181no—
283C22⋊C4—181no—
284C22⋊C4—181no—
285C22⋊C4—181no—
286C22⋊C4—181no—
287C42—181no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—32no—
291C22×C3—31no—
292C22×C3—61no—
293C22×C3—61no—
294C22×C3—61no—
295C22×C3—163no—
296C22×C3#98311yescharacteristic, complemented
297C22×C3#102011yescharacteristic
298C2×D3—121no—
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—483no—
302A4—43no—
303C3×C4—32no—
304C3×C4—32no—
305C3×C4—61no—
306C3×C4—61no—
307Dic3—121no—
308Dic3—121no—
309Dic3—121no—
310Dic3—483no—
311C32—161noSylow
312C23—21no—
313C23—31no—
314C23—31no—
315C23—61no—
316C23—181no—
317C23C2×C3⋊S411yescharacteristic
318Q8—21no—
319Q8—32no—
320Q8—32no—
321Q8—32no—
322Q8—61no—
323Q8—62no—
324Q8—61no—
325Q8—62no—
326Q8—61no—
327Q8—182no—
328Q8C2×C3⋊S412yes—
329D4—182no—
330D4—361no—
331D4—361no—
332D4—362no—
333D4—361no—
334D4—361no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—362no—
351C8—182no—
352C8—182no—
353C2×C3—21no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—61no—
357C2×C3—163no—
358C2×C3—323no—
359C2×C3C2×#102011yescharacteristic
360D3—241no—
361D3—963no—
362C22—31no—
363C22—31no—
364C22—32no—
365C22—31no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—361no—
370C22—361no—
371C22—361no—
372C22#6426411yescharacteristic, complemented
373C22C3⋊#102011yescharacteristic
374C4—32no—
375C4—32no—
376C4—61no—
377C4—61no—
378C4—361no—
379C4—361no—
380C4—361no—
381C3—163no—
382C3#8750711yescharacteristic, complemented
383C2—21no—
384C2—31no—
385C2—31no—
386C2—61no—
387C2—721no—
388C2C2×C3⋊#102011yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊#1020
commutator, cosocle, hypocenterC3×C22⋊SL(2,𝔽3)C22
FittingC23×C3×Q8D3
socleC23×C3C2×S4
hypercenterC22C3⋊#1020
LCSG ▷ C3×C22⋊SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊SL(2,𝔽3)C2semidirect
2. C3⋊#5931C2semidirect
3. C3⋊#5930C2semidirect
9. C23×C3×Q8D3semidirect
10. C2×C22⋊SL(2,𝔽3)D3semidirect
73. C23×Q8C3⋊D3semidirect
102. C2×C3×Q8S4semidirect
252. C2×Q8C3⋊S4semidirect
296. C22×C3#983semidirect
372. C22#64264semidirect
382. C3#87507semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C3⋊#59312
C3⋊#59302
#875073
#840593
#642694
#642644