Finite Group Databasebeta

← #13140890

Subgroups — #13140890

Order 1728.

490 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 93 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×#1084C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3#11721136C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#11720308C211yesmaximal, characteristic
6#11720306C211yesmaximal, characteristic, complemented
7#11720303C211yesmaximal, characteristic, complemented
8#10591566—31nomaximal
9#10591540—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C4⋊#1084—21no—
14C4⋊#1084C2211yescharacteristic
15C4×#1084—21no—
16C4×#1084C2211yescharacteristic
17#3755⋊C3C414yescomplemented
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22C2×#3755D311yescharacteristic
23C3×Q8×Dic3—31no—
24C3×Q8×Dic3—31no—
25C3×Q8×Dic3—61no—
26C2×C3×#215—31no—
27#63894—31no—
28Dic3×D12—31no—
29#63662D311yescharacteristic, complemented
30C3×#835—31no—
31#63576—31no—
32#63575—31no—
33#63564—31no—
34#63562—31no—
35#63554—31no—
36#63554—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×#1084D411yescharacteristic
44C2×#1084D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yes—
48((C3⋊D3)⋊C3)⋊C4—42no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51#5164—91no—
52#5026—91no—
53#4817—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)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×#215—34no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4⋊Dic3—61no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3—62no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—31no—
75C3×C4×Dic3—61no—
76C3×C4×Dic3—61no—
77C3×C4×Dic3—61no—
78C3×C4×Dic3—62no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81#3722—61no—
82(C2×D3)⋊Dic3—61no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×(C3⋊D3)⋊C3—22no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—22no—
88C4×H(3,𝔽3)C22×C411yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90#1084—21no—
91#1084—21no—
92#1084—41no—
93(C3⋊Dic3)⋊C3—41no—
94C2×C3×SD16—91no—
95C3×C4×Q8—91no—
96Q8×Dic3—31no—
97Q8×Dic3—31no—
98Q8×Dic3—61no—
99C2×#215—31no—
100C2×#215—91no—
101D4×Dic3—91no—
102#893—91no—
103C4×D12—91no—
104C3×#103—91no—
105#835—31no—
106#835—91no—
107#832—91no—
108#821—91no—
109#818—91no—
110#815—91no—
111#815—91no—
112#809—91no—
113#807—91no—
114#807—91no—
115#803—91no—
116C22×C3⋊D3—21no—
117C22×C3×D3—61no—
118C32×Q8—22no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—22no—
122C32×Q8C4×D312yescomplemented
123C32×Q8C2×Dic312yescomplemented
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)C2×Dic312yes—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3×D12—32no—
144C3×D12—61no—
145#624—122no—
146D3×Dic3—122no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)C2×Dic312yescomplemented
149C3×C3⋊C8—32no—
150C3×C3⋊C8—61no—
151#440—271noSylow
152C2×H(3,𝔽3)C4×D411yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154C2×H(3,𝔽3)#13711yescharacteristic
155(C3⋊D3)⋊C3—42no—
156C2×C3×Q8—31no—
157C2×C3×Q8—31no—
158C2×C3×Q8—61no—
159C2×C3×Q8D3211yescharacteristic
160C2×C3×D4—91no—
161C22×Dic3—181no—
162C2×D12—31no—
163C2×D12—91no—
164C2×C4×D3—181no—
165C2×C4×D3—181no—
166C3×SD16—94no—
167C2×C3×C8—91no—
168C2×C3×C8—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4—181no—
172C3×C42—91no—
173C3×C42—181no—
174#217—181no—
175#215—34no—
176#215—94no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—61no—
183C4⋊Dic3—61no—
184C4⋊Dic3—61no—
185C4⋊Dic3—62no—
186C4⋊Dic3—91no—
187C4×Dic3—31no—
188C4×Dic3—31no—
189C4×Dic3—61no—
190C4×Dic3—61no—
191C4×Dic3—61no—
192C4×Dic3—62no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—91no—
196C2×C3⋊C8—91no—
197C22×C32—21no—
198C22×C32D3×D411yescharacteristic
199C22×C32D3×D411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—62no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C32×C4—24no—
207C32×C4—22no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4C2×C4×D311yescharacteristic
212C32×C4#23711yescharacteristic
213C32×C4#23711yescharacteristic
214C32×C4C22×Dic311yescharacteristic
215C3⋊Dic3—41no—
216C3×Dic3—62no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—121no—
222C3×Dic3—122no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C2×SD16—91no—
227C4×Q8—91no—
228C4×D4—271no—
229#107—271no—
230#103—91no—
231#102—271no—
232#97—271no—
233H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
234C23×C3—181no—
235C22×D3—61no—
236C22×D3—181no—
237C3×Q8—21no—
238C3×Q8—32no—
239C3×Q8—32no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×Q8—62no—
243C3×Q8—62no—
244C3×Q8D3×Dic312yes—
245C3×D4—92no—
246C3×D4—181no—
247C2×C3×C4—21no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—62no—
253C2×C3×C4—61no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4C2×D3211yescharacteristic
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—62no—
268C2×Dic3—61no—
269C2×Dic3—181no—
270C2×Dic3—362no—
271D12—32no—
272D12—61no—
273D12—92no—
274D12—181no—
275C4×D3—362no—
276C4×D3—362no—
277C3×C8—92no—
278C3×C8—181no—
279C3⋊C8—32no—
280C3⋊C8—61no—
281C3⋊C8—92no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32#67⋊C411yescharacteristic
287C2×C32#91411yescharacteristic
288C2×C32#91411yescharacteristic
289C2×C32#91511yescharacteristic
290C2×C32#91811yescharacteristic
291C2×C32D4×Dic311yescharacteristic
292C3⋊D3—42no—
293C3×D3—122no—
294C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C2×D4—91no—
296C22×C4—541no—
297SD16—94no—
298C2×C8—91no—
299C2×C8—91no—
300C4⋊C4—91no—
301C4⋊C4—91no—
302C4⋊C4—181no—
303C22⋊C4—541no—
304C42—91no—
305C42—181no—
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—61no—
309C22×C3—182no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3#381111yescharacteristic
313C2×D3—62no—
314C2×D3—121no—
315C2×D3—121no—
316C2×D3—182no—
317C2×D3—361no—
318C2×D3—361no—
319C3×C4—22no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—62no—
325C3×C4—62no—
326C3×C4—64no—
327C3×C4—61no—
328C3×C4—61no—
329C3×C4—181no—
330C3×C4—181no—
331C3×C4—361no—
332C3×C4—361no—
333C3×C4C2×D3×Dic311yescharacteristic
334C3×C4#380511yescharacteristic
335Dic3—61no—
336Dic3—61no—
337Dic3—61no—
338Dic3—61no—
339Dic3—62no—
340Dic3—121no—
341Dic3—121no—
342Dic3—121no—
343Dic3—122no—
344Dic3—121no—
345Dic3—361no—
346C32—21no—
347C32#481711yescharacteristic, complemented
348C32#516411yescharacteristic, complemented
349C23—181no—
350Q8—21no—
351Q8((C3⋊D3)⋊C3)⋊C412yescomplemented
352D4—92no—
353D4—181no—
354C2×C4—21no—
355C2×C4—91no—
356C2×C4—91no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—1082no—
360C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C8—92no—
362C8—181no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—362no—
373C2×C3#6394111yescharacteristic
374C2×C3#6395111yescharacteristic
375C2×C3#67⋊Dic311yescharacteristic
376D3—122no—
377D3—362no—
378C22—182no—
379C22—361no—
380C22—361no—
381C22#8867711yescharacteristic
382C4—22no—
383C4—181no—
384C4—181no—
385C4—361no—
386C4—361no—
387C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
388C4#8867211yescharacteristic
389C3—31no—
390C3—31no—
391C3—61no—
392C3#1059157211yescharacteristic
393C2—362no—
394C2#1172122911yescharacteristic
395C2#1172123111yescharacteristic
396C2#1172131511yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((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×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1084C2semidirect
4. #63662⋊C3C2semidirect
6. #11720306C2semidirect
7. #11720303C2semidirect
17. #3755⋊C3C4semidirect
29. #63662D3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
61. #3755Dic3semidirect
122. C32×Q8C4×D3semidirect
123. C32×Q8C2×Dic3semidirect
148. C3⋊(C3⋊C8)C2×Dic3semidirect
233. H(3,𝔽3)#440semidirect
347. C32#4817semidirect
348. C32#5164semidirect
351. 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×#10842
C2×#3755⋊C32
#117211362
#63662⋊C32
#117203082
#117203062
#117203032
#105915663
#105915403