Finite Group Databasebeta

← #10592516⋊C3 #13141928

Subgroups — #10592516⋊C3 #13141928

Order 1728.

462 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 112 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#64102⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592516C311yesmaximal, characteristic, complemented
9C3×#4817—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3755⋊C3C414yescomplemented
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
23C2×#3755C2×C311yescharacteristic
24#64102C2×C311yescharacteristic
25C3×Q8×Dic3—31no—
26C2×C3×#215—31no—
27C3×#893—31no—
28C32×#103—21no—
29C32×#103D311yescharacteristic, complemented
30C32×#103C2×C311yescharacteristic, complemented
31#63648C2×C311yescharacteristic
32#63642C2×C311yescharacteristic, complemented
33#63634C2×C311yescharacteristic, complemented
34C3×#821—31no—
35C3×#815—31no—
36C3×#807—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)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yes—
48C4×(C3⋊D3)⋊C3—42no—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51C3×#440—91no—
52#4817—31no—
53#4817—91no—
54C2×C32×Q8—21no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×Q8C22×C311yescharacteristic
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×C3×D12—31no—
60C2×C3×C4×D3—61no—
61C2×C32×C8—21no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C22×C311yescharacteristic
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67#3755C3×C414yescomplemented
68#3752—21no—
69C4⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C4×C3⋊Dic3—21no—
72C4×C3⋊Dic3C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×#215—34no—
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—61no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—61no—
80C2×C3×C3⋊C8—31no—
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)—41no—
87C4×H(3,𝔽3)C22×C411yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92C2×C3×SD16—91no—
93C3×C4×Q8—91no—
94C3×C4×D4—91no—
95Q8×Dic3—31no—
96Q8×Dic3—91no—
97C2×#215—31no—
98C2×#215—91no—
99#893—31no—
100#893—91no—
101C3×#107—91no—
102C3×#103—31no—
103C3×#103—31no—
104C3×#103—61no—
105C3×#103C3×D311yescharacteristic
106C3×#102—91no—
107C3×#97—91no—
108#821—31no—
109#821—91no—
110#815—31no—
111#815—91no—
112#807—31no—
113#807—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×C3×C412yescomplemented
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C23×C311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3C3×D411yescharacteristic
133C2×C3⋊Dic3C3×D411yescharacteristic
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)C2×C3×C412yes—
136C4×C3⋊D3—42no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C3×D12—32no—
141C3×D12—61no—
142C3×C4×D3—122no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—41no—
146C3⋊(C3⋊C8)C2×C3×C412yescomplemented
147C3×C3⋊C8—32no—
148#440—91noSylow
149C2×H(3,𝔽3)C4×D411yescharacteristic
150C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
151C2×H(3,𝔽3)#13711yescharacteristic
152(C3⋊D3)⋊C3—42no—
153C2×C3×Q8—31no—
154C2×C3×Q8—31no—
155C2×C3×Q8—61no—
156C2×C3×Q8C2×C3×D311yescharacteristic
157C2×C3×D4—91no—
158C22×C3×C4—181no—
159C2×D12—31no—
160C2×D12—91no—
161C2×C4×D3—61no—
162C2×C4×D3—181no—
163C3×SD16—94no—
164C2×C3×C8—31no—
165C2×C3×C8—31no—
166C2×C3×C8—61no—
167C2×C3×C8—91no—
168C2×C3×C8C2×C3×D311yescharacteristic
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—181no—
174C3×C4⋊C4C2×C3×D311yescharacteristic
175C3×C22⋊C4—181no—
176C3×C42—91no—
177C3×C42—181no—
178#215—34no—
179#215—94no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—31no—
183C4⋊Dic3—61no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186C4×Dic3—31no—
187C4×Dic3—61no—
188C4×Dic3—91no—
189C4×Dic3—181no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×C3×D411yescharacteristic
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×C4—41no—
207C32×C4—41no—
208C32×C4—81no—
209C32×C4C2×C4×D311yescharacteristic
210C32×C4#23711yescharacteristic
211C32×C4C22×C3×C411yescharacteristic
212C32×C4C3×#4111yescharacteristic
213C3⋊Dic3—21no—
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—121no—
219C2×SD16—91no—
220C4×Q8—91no—
221C4×D4—91no—
222#107—91no—
223#103(C3⋊D3)⋊C311yescharacteristic, complemented
224#102—91no—
225#97—91no—
226H(3,𝔽3)#44011yescharacteristic, 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×Q8C3×C4×D312yes—
238C3×D4—92no—
239C3×D4—181no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4—181no—
254C2×C3×C4—362no—
255C2×C3×C4C22×C3×D311yescharacteristic
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—61no—
259C2×Dic3—91no—
260C2×Dic3—91no—
261C2×Dic3—181no—
262D12—32no—
263D12—61no—
264D12—92no—
265D12—181no—
266C4×D3—122no—
267C4×D3—362no—
268C3×C8—21no—
269C3×C8—61no—
270C3×C8—61no—
271C3×C8—92no—
272C3×C8—121no—
273C3⋊C8—32no—
274C3⋊C8—92no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32#67⋊C411yescharacteristic
279C2×C32#91411yescharacteristic
280C2×C32#91811yescharacteristic
281C2×C32C3×C4×D411yescharacteristic
282C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
283C2×C32C3×#13711yescharacteristic
284C3⋊D3—42no—
285C3×D3—122no—
286C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
287C2×D4—91no—
288C22×C4—181no—
289SD16—94no—
290C2×C8—91no—
291C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
292C4⋊C4—91no—
293C4⋊C4—181no—
294C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
295C22⋊C4—181no—
296C42—91no—
297C42—181no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—61no—
301C22×C3—182no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3C3×D3×D411yescharacteristic
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—41no—
317C3×C4—62no—
318C3×C4—62no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—122no—
322C3×C4—121no—
323C3×C4—121no—
324C3×C4—181no—
325C3×C4—181no—
326C3×C4—241no—
327C3×C4—361no—
328C3×C4C2×C3×C4×D311yescharacteristic
329C3×C4C3×#23711yescharacteristic
330Dic3—61no—
331Dic3—61no—
332Dic3—121no—
333Dic3—181no—
334Dic3—181no—
335Dic3—361no—
336C32—21no—
337C32#481711yescharacteristic, complemented
338C32C3×#44011yescharacteristic, complemented
339C23—181no—
340Q8—21no—
341Q8C4×(C3⋊D3)⋊C312yescomplemented
342D4—92no—
343D4—181no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—91no—
347C2×C4—91no—
348C2×C4—181no—
349C2×C4—362no—
350C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
351C8—21no—
352C8—92no—
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×C3C3×#67⋊C411yescharacteristic
364C2×C3C3×#91411yescharacteristic
365C2×C3C3×#91811yescharacteristic
366D3—122no—
367D3—362no—
368C22—182no—
369C22—361no—
370C22—361no—
371C22D4×(C3⋊D3)⋊C311yescharacteristic
372C4—22no—
373C4—41no—
374C4—181no—
375C4—181no—
376C4—361no—
377C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
378C4#3830⋊C311yescharacteristic
379C3—31no—
380C3—31no—
381C3—61no—
382C3C3×#481711yescharacteristic
383C2—362no—
384C2#64090⋊C311yescharacteristic
385C2#64123⋊C311yescharacteristic
386C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C32×C4C23×C3
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#440
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. H(3,𝔽3)×#103C2semidirect
6. #63642⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592516C3semidirect
15. #3755⋊C3C4semidirect
22. Q8×C3⋊Dic3C2×C3semidirect
29. C32×#103D3semidirect
30. C32×#103C2×C3semidirect
32. #63642C2×C3semidirect
33. #63634C2×C3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
67. #3755C3×C4semidirect
120. C32×Q8C4×D3semidirect
121. C32×Q8C2×C3×C4semidirect
146. C3⋊(C3⋊C8)C2×C3×C4semidirect
223. #103(C3⋊D3)⋊C3semidirect
226. H(3,𝔽3)#440semidirect
337. C32#4817semidirect
338. C32C3×#440semidirect
341. Q8C4×(C3⋊D3)⋊C3semidirect

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
#64102⋊C32
H(3,𝔽3)×#1032
#63648⋊C32
#63642⋊C32
#63634⋊C32
#105925163
C3×#48173