Finite Group Databasebeta

← #13138952

Subgroups — #13138952

Order 1728.

470 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721149C211yesmaximal, characteristic
2#11721082C211yesmaximal, characteristic, complemented
3C2×#88437C211yesmaximal, characteristic
4#11720554C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590747—31nomaximal
9#10590743—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#6199—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88451—21no—
19#88448—21no—
20#88447—21no—
21#88437C414yescomplemented
22#63898—31no—
23Dic3×D12—31no—
24#63844—31no—
25#63843—31no—
26C2×#3716—31no—
27C2×#3715—31no—
28#63643D311yescharacteristic, complemented
29#63642D311yescharacteristic, complemented
30C3×#821—31no—
31C3×#821—31no—
32C3×#821—61no—
33C3×#816—31no—
34C3×#815—31no—
35#63576—31no—
36#63575—31no—
37C22×#272—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
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—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51#4817—91no—
52#4757—91no—
53#4710—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—
61C2×C32×C8—21no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C3×C4×Dic3—31no—
72C2×C3×C3⋊C8—31no—
73#3724—61no—
74Dic3⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77Dic32—61no—
78Dic32—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)C22×C411yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—41no—
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92Q8×Dic3—91no—
93C2×#215—91no—
94D4×Dic3—91no—
95C2×#214—91no—
96C2×#204—91no—
97#893—91no—
98#887—91no—
99C4×D12—91no—
100C4×Dic6—91no—
101C3×#103—91no—
102C3×#102—91no—
103C3×#97—91no—
104#821—31no—
105#821—31no—
106#821—61no—
107#818—91no—
108#816—31no—
109#816—91no—
110#815—31no—
111#815—91no—
112#807—91no—
113#807—91no—
114C22×C3×D3—61no—
115C22×C3×D3—61no—
116C22×C3×D3—121no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3D1211yescharacteristic
123C2×C3⋊Dic3D1211yescharacteristic
124#634—21no—
125#634C4×D312yescomplemented
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—121no—
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—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—41no—
146C3⋊(C3⋊C8)C4×D312yescomplemented
147C3×C3⋊C8—32no—
148#440—271noSylow
149C2×H(3,𝔽3)C4×D411yescharacteristic
150C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
151C2×H(3,𝔽3)#13711yescharacteristic
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—31no—
164C2×C3×C8—31no—
165C2×C3×C8—61no—
166C2×C3×C8—91no—
167C2×C3×C8D3211yescharacteristic
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C42—91no—
171#217—181no—
172#215—94no—
173#214—94no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—61no—
180C4⋊Dic3—91no—
181C4⋊Dic3—181no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184C4×Dic3—31no—
185C4×Dic3—91no—
186C4×Dic3—181no—
187C4×Dic3—181no—
188C4×Dic3—181no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—91no—
191#204—94no—
192C22×C32—21no—
193C22×C32C2×D1211yescharacteristic
194C22×C32D3×D411yescharacteristic
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×C4C2×C4×D311yescharacteristic
207C32×C4C2×C4×D311yescharacteristic
208C32×C4#23511yescharacteristic
209C32×C4#23711yescharacteristic
210C3⋊Dic3—22no—
211C3⋊Dic3—21no—
212C3⋊Dic3—21no—
213C3⋊Dic3—41no—
214C3×Dic3—62no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—241no—
221C2×SD16—271no—
222C4×Q8—271no—
223C4×D4—271no—
224#107—271no—
225#103—91no—
226#102—91no—
227#97—91no—
228H(3,𝔽3)#44011yescharacteristic, 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—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4C2×D3211yescharacteristic
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—121no—
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—21no—
271C3×C8—61no—
272C3×C8—61no—
273C3×C8—92no—
274C3×C8—121no—
275C3⋊C8—32no—
276C3⋊C8—92no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32C4×D1211yescharacteristic
281C2×C32#67⋊C411yescharacteristic
282C2×C32#90811yescharacteristic
283C2×C32#90911yescharacteristic
284C2×C32#91411yescharacteristic
285C2×C32#91811yescharacteristic
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((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C4⋊C4—91no—
296C4⋊C4—91no—
297C4⋊C4—541no—
298C22⋊C4—541no—
299C42—91no—
300C42—541no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—61no—
304C22×C3—182no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3D3×D1211yescharacteristic
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—61no—
322C3×C4—61no—
323C3×C4—181no—
324C3×C4—181no—
325C3×C4—182no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4C4×D3211yescharacteristic
329C3×C4#380411yescharacteristic
330Dic3—61no—
331Dic3—61no—
332Dic3—62no—
333Dic3—121no—
334Dic3—121no—
335Dic3—121no—
336Dic3—182no—
337Dic3—181no—
338Dic3—181no—
339Dic3—241no—
340Dic3—361no—
341C32—21no—
342C32#471011yescharacteristic, complemented
343C32#481711yescharacteristic, complemented
344C23—181no—
345Q8—92no—
346Q8—181no—
347D4—92no—
348D4—181no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—1082no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C8—21no—
357C8—92no—
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#6379411yescharacteristic
369C2×C3#6380111yescharacteristic
370C2×C3#6388011yescharacteristic
371D3—122no—
372D3—122no—
373D3—242no—
374C22—182no—
375C22—361no—
376C22—361no—
377C22#8865411yescharacteristic
378C4—181no—
379C4—181no—
380C4—182no—
381C4—361no—
382C4—361no—
383C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C4#8867111yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3#1059079511yescharacteristic
389C2—362no—
390C2#1172084811yescharacteristic
391C2#1172085311yescharacteristic
392C2#1172112311yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×C4⋊#272C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#440
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721082C2semidirect
5. #63643⋊C3C2semidirect
6. #63642⋊C3C2semidirect
7. #11720306C2semidirect
21. #88437C4semidirect
28. #63643D3semidirect
29. #63642D3semidirect
47. #634⋊C3C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
125. #634C4×D3semidirect
146. C3⋊(C3⋊C8)C4×D3semidirect
228. H(3,𝔽3)#440semidirect
342. C32#4710semidirect
343. C32#4817semidirect

Maximal subgroups

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

subgroupindex
#117211492
#117210822
C2×#884372
#117205542
#63643⋊C32
#63642⋊C32
#117203062
#105907473
#105907433