Finite Group Databasebeta

← #13140911

Subgroups — #13140911

Order 1728.

469 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 99 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
1Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3#11721132C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic
6#11720310C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591571—31nomaximal
9#10591536—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3755⋊C3C414yescomplemented
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22Q8×C3⋊Dic3D311yescharacteristic, complemented
23C2×#3755D311yescharacteristic
24C3×Q8×Dic3—31no—
25C2×C3×#215—31no—
26#63898—31no—
27#63894—31no—
28C3×#835—31no—
29C3×#835—31no—
30C3×#835—61no—
31#63574—31no—
32#63573—31no—
33#63564—31no—
34#63563—31no—
35#63553—31no—
36Dic3×C3⋊C8—31no—
37C22×(C3⋊D3)⋊C3—21no—
38Q8×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)C2×C412yescomplemented
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yes—
48((C3⋊D3)⋊C3)⋊C4—42no—
49#6180—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51Dic3×SD16—91no—
52C4×#215—91no—
53#4788—91no—
54C2×C32×Q8—21no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×Q8C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×#624—61no—
60C2×D3×Dic3—61no—
61#3755Dic314yescomplemented
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4×C3⋊Dic3—21no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×#215—34no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3—61no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—61no—
79#3722—61no—
80(C2×D3)⋊Dic3—61no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—41no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3SD1612yescomplemented
92C2×C3×SD16—91no—
93C3×C4×Q8—91no—
94Q8×Dic3—31no—
95Q8×Dic3—91no—
96C2×#215—31no—
97C2×#215—91no—
98D4×Dic3—91no—
99#893—91no—
100C4×D12—91no—
101C3×#103—91no—
102#835—31no—
103#835—31no—
104#835—61no—
105#832—91no—
106#821—91no—
107#817—91no—
108C8×Dic3—91no—
109C8×Dic3—91no—
110#809—91no—
111#808—91no—
112#808—91no—
113C4×C3⋊C8—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—22no—
119C32×Q8—41no—
120C32×Q8C4×D312yescomplemented
121C32×Q8C2×Dic312yescomplemented
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3#6711yescharacteristic
130C2×C3⋊Dic3#6711yescharacteristic
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)C2×Dic312yes—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—121no—
139C3×D12—32no—
140C3×D12—61no—
141#624—122no—
142D3×Dic3—122no—
143C3⋊(C3⋊C8)C2×Dic312yescomplemented
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(C3⋊D3)⋊C3—42no—
153C2×C3×Q8—31no—
154C2×C3×Q8—31no—
155C2×C3×Q8—61no—
156C2×C3×Q8D3211yescharacteristic
157C2×C3×D4—91no—
158C22×Dic3—181no—
159C2×D12—31no—
160C2×D12—91no—
161C2×C4×D3—181no—
162C2×C4×D3—181no—
163C3×SD16—94no—
164C2×C3×C8—91no—
165C2×C3×C8—91no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4—181no—
169C3×C42—91no—
170C3×C42—181no—
171#217—181no—
172#215—34no—
173#215—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—61no—
181C4⋊Dic3—91no—
182C4⋊Dic3—181no—
183C4×Dic3—31no—
184C4×Dic3—61no—
185C4×Dic3—91no—
186C4×Dic3—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×#6711yescharacteristic
195C2×C3⋊D3—22no—
196C2×C3⋊D3—41no—
197C2×C3⋊D3—41no—
198C2×C3×D3—62no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C32×C4—22no—
202C32×C4—22no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—42no—
206C32×C4C2×C4×D311yescharacteristic
207C32×C4#23511yescharacteristic
208C32×C4#23711yescharacteristic
209C32×C4C22×Dic311yescharacteristic
210C3⋊Dic3—21no—
211C3⋊Dic3—41no—
212C3⋊Dic3#21512yescomplemented
213C3×Dic3—32no—
214C3×Dic3—61no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—241no—
219C2×SD16—91no—
220C4×Q8—91no—
221C4×D4—271no—
222#106—271no—
223#103—91no—
224#102—271no—
225C4×C8—271no—
226H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—61no—
229C22×D3—181no—
230C3×Q8—21no—
231C3×Q8—32no—
232C3×Q8—32no—
233C3×Q8—61no—
234C3×Q8—61no—
235C3×Q8—62no—
236C3×Q8—121no—
237C3×Q8D3×Dic312yes—
238C3×D4—92no—
239C3×D4—181no—
240C2×C3×C4—21no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—121no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3211yescharacteristic
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—91no—
258C2×Dic3—91no—
259C2×Dic3—121no—
260C2×Dic3—181no—
261C2×Dic3—362no—
262D12—32no—
263D12—61no—
264D12—92no—
265D12—181no—
266C4×D3—362no—
267C4×D3—362no—
268C3×C8—92no—
269C3×C8—181no—
270C3⋊C8—32no—
271C3⋊C8—61no—
272C3⋊C8—61no—
273C3⋊C8—92no—
274C3⋊C8—121no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32D3×SD1611yescharacteristic
279C2×C32#91611yescharacteristic
280C2×C32C4×#6711yescharacteristic
281C2×C32D4×Dic311yescharacteristic
282C2×C32C2×#21511yescharacteristic
283C2×C32#95011yescharacteristic
284C3⋊D3—42no—
285C3×D3—122no—
286C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
287C2×D4—91no—
288C22×C4—541no—
289SD16—94no—
290C2×C8—91no—
291C2×C8—91no—
292C4⋊C4—91no—
293C4⋊C4—91no—
294C4⋊C4—181no—
295C22⋊C4—541no—
296C42—91no—
297C42—181no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—61no—
301C22×C3—182no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3D3×#6711yescharacteristic
305C2×D3—62no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—182no—
309C2×D3—361no—
310C2×D3—361no—
311C3×C4—22no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—62no—
317C3×C4—62no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—92no—
321C3×C4—122no—
322C3×C4—181no—
323C3×C4—361no—
324C3×C4—361no—
325C3×C4C2×D3×Dic311yescharacteristic
326C3×C4#380411yescharacteristic
327Dic3—32no—
328Dic3—61no—
329Dic3—92no—
330Dic3—121no—
331Dic3—121no—
332Dic3—121no—
333Dic3—181no—
334Dic3—241no—
335Dic3—361no—
336C32—21no—
337C32C4×#21511yescharacteristic, complemented
338C32Dic3×SD1611yescharacteristic, complemented
339C23—181no—
340Q8—21no—
341Q8((C3⋊D3)⋊C3)⋊C412yescomplemented
342D4—92no—
343D4—181no—
344C2×C4—21no—
345C2×C4—91no—
346C2×C4—91no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—1082no—
350C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C8—92no—
352C8—181no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—362no—
363C2×C3D3×#21511yescharacteristic
364C2×C3#6394711yescharacteristic
365C2×C3Dic3×#6711yescharacteristic
366D3—122no—
367D3—362no—
368C22—182no—
369C22—361no—
370C22—361no—
371C22#8867611yescharacteristic
372C4—22no—
373C4—92no—
374C4—181no—
375C4—361no—
376C4—361no—
377C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
378C4#8867111yescharacteristic
379C3—31no—
380C3—31no—
381C3—61no—
382C3Dic3×#21511yescharacteristic
383C2—362no—
384C2#1172123211yescharacteristic
385C2#1172123911yescharacteristic
386C2#1172131411yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. #11720583C2semidirect
6. #11720310C2semidirect
7. #11720307C2semidirect
15. #3755⋊C3C4semidirect
22. Q8×C3⋊Dic3D3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
61. #3755Dic3semidirect
91. (C3⋊Dic3)⋊C3SD16semidirect
120. C32×Q8C4×D3semidirect
121. C32×Q8C2×Dic3semidirect
143. C3⋊(C3⋊C8)C2×Dic3semidirect
212. C3⋊Dic3#215semidirect
226. H(3,𝔽3)C4×SD16semidirect
337. C32C4×#215semidirect
338. C32Dic3×SD16semidirect
341. Q8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
Q8×(C3⋊Dic3)⋊C32
C2×#3755⋊C32
#117211322
#117205832
#117203182
#117203102
#117203072
#105915713
#105915363