Finite Group Databasebeta

← #13137990

Subgroups — #13137990

Order 1728.

477 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#64096⋊C3C211yesmaximal, characteristic, complemented
2#11721149C211yesmaximal, characteristic
3C2×#88437C211yesmaximal, characteristic
4#11720531C211yesmaximal, characteristic
5#63635⋊C3C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10590460—31nomaximal
9#10590453—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#6199—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#88437C414yescomplemented
22#64096D311yescharacteristic, complemented
23C3×#887—31no—
24#63898—31no—
25Dic3×D12—31no—
26C2×#3716—31no—
27C2×#3715—31no—
28#63635D311yescharacteristic, complemented
29C3×#809—31no—
30C3×#809—31no—
31C3×#809—61no—
32C3×#808—31no—
33#63567—31no—
34#63566—31no—
35#63553—31no—
36Dic3×C3⋊C8—31no—
37C22×#272—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C4⋊#272—21no—
41C4⋊#272C2×C412yes—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46#634⋊C3—21no—
47#634⋊C3C2×C412yescomplemented
48#6199—42no—
49#6180—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51C4×#215—91no—
52#4852—91no—
53#4758—91no—
54C2×#634C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×D12—31no—
57C2×C3×D12—61no—
58C2×C3×Dic6—31no—
59C2×D3×Dic3—61no—
60C2×D3×Dic3—61no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C4⋊C4C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—21no—
66C4×C3⋊Dic3—21no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×Dic3⋊C4—61no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77(C2×D3)⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79#3716—34no—
80#3715—34no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×#272—22no—
83C2×#272—41no—
84C2×#272—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3SD1612yescomplemented
92C3×C4×Q8—91no—
93C2×#215—91no—
94D4×Dic3—91no—
95C2×#214—91no—
96C2×#204—91no—
97#893—91no—
98#887—31no—
99#887—91no—
100C4×D12—91no—
101C3×#106—91no—
102C3×#102—91no—
103#835—91no—
104#816—91no—
105C8×Dic3—91no—
106C8×Dic3—91no—
107#810—91no—
108#809—31no—
109#809—31no—
110#809—61no—
111#808—31no—
112#808—91no—
113C4×C3⋊C8—91no—
114C22×C3×D3—61no—
115C22×C3×D3—61no—
116C22×C3×D3—121no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3⋊Dic3#6711yescharacteristic
127#634—21no—
128#634C4×D312yescomplemented
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×D12—32no—
134C3×D12—32no—
135C3×D12—62no—
136C3×D12—61no—
137C3×D12—61no—
138C3×D12—121no—
139C3×Dic6—32no—
140C3×Dic6—61no—
141D3×Dic3—122no—
142D3×Dic3—122no—
143C3⋊(C3⋊C8)C4×D312yescomplemented
144C3×C3⋊C8—32no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147C3×C3⋊C8—121no—
148C4×SD16—271noSylow
149C2×H(3,𝔽3)C4×D411yescharacteristic
150C2×H(3,𝔽3)C2×SD1611yescharacteristic
151C2×H(3,𝔽3)#13511yescharacteristic
152#272—42no—
153C2×C3×Q8—91no—
154C2×C3×D4—91no—
155C22×Dic3—181no—
156C2×D12—31no—
157C2×D12—31no—
158C2×D12—61no—
159C2×C4×D3—181no—
160C2×C4×D3—181no—
161C2×Dic6—31no—
162C2×Dic6—91no—
163C2×C3×C8—91no—
164C2×C3×C8—91no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4D3211yescharacteristic
171C3×C42—91no—
172C3×C42—181no—
173#217—181no—
174#215—94no—
175#214—94no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—31no—
179C4⋊Dic3—91no—
180Dic3⋊C4—61no—
181Dic3⋊C4—181no—
182C4×Dic3—31no—
183C4×Dic3—61no—
184C4×Dic3—91no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—61no—
190C2×C3⋊C8—91no—
191#204—94no—
192C22×C32—21no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×#6711yescharacteristic
195C2×C3×D3—62no—
196C2×C3×D3—62no—
197C2×C3×D3—122no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C2×C3×D3—241no—
203C2×C3×D3—241no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—41no—
207C32×C4—41no—
208C32×C4—81no—
209C32×C4C2×C4×D311yescharacteristic
210C32×C4C2×C4×D311yescharacteristic
211C32×C4#23511yescharacteristic
212C32×C4#23911yescharacteristic
213C3⋊Dic3—22no—
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3⋊Dic3#21512yescomplemented
217C3×Dic3—32no—
218C3×Dic3—62no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C2×SD16—271no—
222C4×Q8—91no—
223C4×D4—271no—
224#106—91no—
225#103—271no—
226#102—91no—
227C4×C8—271no—
228H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
229C23×C3—181no—
230C22×D3—61no—
231C22×D3—61no—
232C22×D3—121no—
233C3×Q8—92no—
234C3×Q8—181no—
235C3×D4—92no—
236C3×D4—181no—
237C2×C3×C4—21no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—121no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4C2×D3211yescharacteristic
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—91no—
254C2×Dic3—91no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C2×Dic3—362no—
258D12—32no—
259D12—32no—
260D12—62no—
261D12—61no—
262D12—61no—
263D12—121no—
264C4×D3—362no—
265C4×D3—362no—
266Dic6—32no—
267Dic6—61no—
268Dic6—92no—
269Dic6—181no—
270C3×C8—92no—
271C3×C8—181no—
272C3⋊C8—32no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—92no—
276C3⋊C8—121no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32#89311yescharacteristic
281C2×C32D3×SD1611yescharacteristic
282C2×C32#91611yescharacteristic
283C2×C32C4×#6711yescharacteristic
284C2×C32C2×#21511yescharacteristic
285C2×C32#95011yescharacteristic
286C3×D3—122no—
287C3×D3—122no—
288C3×D3—242no—
289C2×Q8—91no—
290C2×D4—91no—
291C22×C4—541no—
292SD16—274no—
293C2×C8—91no—
294C2×C8—91no—
295C4⋊C4—91no—
296C4⋊C4—181no—
297C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C22⋊C4—541no—
299C42—91no—
300C42—181no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—61no—
304C22×C3—182no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3D3×#6711yescharacteristic
308C2×D3—62no—
309C2×D3—62no—
310C2×D3—122no—
311C2×D3—121no—
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—121no—
315C2×D3—241no—
316C2×D3—241no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—41no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—92no—
325C3×C4—121no—
326C3×C4—121no—
327C3×C4—181no—
328C3×C4—182no—
329C3×C4—241no—
330C3×C4—361no—
331C3×C4#379911yescharacteristic
332C3×C4C4×D3211yescharacteristic
333Dic3—32no—
334Dic3—62no—
335Dic3—61no—
336Dic3—92no—
337Dic3—121no—
338Dic3—182no—
339Dic3—181no—
340Dic3—361no—
341C32—21no—
342C32#485211yescharacteristic, complemented
343C32C4×#21511yescharacteristic, complemented
344C23—181no—
345Q8—92no—
346Q8—181no—
347D4—92no—
348D4—181no—
349C2×C4—21no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—1082no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C8—92no—
357C8—181no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—362no—
368C2×C3#6390411yescharacteristic
369C2×C3#6393511yescharacteristic
370C2×C3D3×#21511yescharacteristic
371D3—122no—
372D3—122no—
373D3—242no—
374C22—182no—
375C22—361no—
376C22—361no—
377C22#8867611yescharacteristic
378C4—41no—
379C4—92no—
380C4—182no—
381C4—181no—
382C4—361no—
383C4#8865211yescharacteristic
384C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3#1059045811yescharacteristic
389C2—362no—
390C2#1172115411yescharacteristic
391C2#1172119711yescharacteristic
392C2#1172123211yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊#272C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64096⋊C3C2semidirect
5. #63635⋊C3C2semidirect
6. #11720313C2semidirect
7. #11720310C2semidirect
21. #88437C4semidirect
22. #64096D3semidirect
28. #63635D3semidirect
47. #634⋊C3C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
91. (C3⋊Dic3)⋊C3SD16semidirect
128. #634C4×D3semidirect
143. C3⋊(C3⋊C8)C4×D3semidirect
216. C3⋊Dic3#215semidirect
228. H(3,𝔽3)C4×SD16semidirect
342. C32#4852semidirect
343. C32C4×#215semidirect

Maximal subgroups

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

subgroupindex
#64096⋊C32
#117211492
C2×#884372
#117205312
#63635⋊C32
#117203132
#117203102
#105904603
#105904533