Finite Group Databasebeta

← #10592490⋊C3 #13141901

Subgroups — #10592490⋊C3 #13141901

Order 1728.

378 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 100 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
2#64097⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#63644⋊C3C211yesmaximal, characteristic
6#63641⋊C3C211yesmaximal, characteristic
7#63635⋊C3C211yesmaximal, characteristic
8#10592490C311yesmaximal, characteristic, complemented
9C3×#4791—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21Q8×C3⋊Dic3C2×C311yescharacteristic
22#64097C2×C311yescharacteristic
23C3×Q8×Dic3—31no—
24C3×#888—31no—
25C32×#103—21no—
26C32×#103D311yescharacteristic
27C32×#103C2×C311yescharacteristic
28#63662C2×C311yescharacteristic
29#63644C2×C311yescharacteristic
30#63641C2×C311yescharacteristic
31#63635C2×C311yescharacteristic
32C3×#835—31no—
33C3×#817—31no—
34C3×Dic3⋊C8—31no—
35C3×#808—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)Q812yescomplemented
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#634⋊C3—42no—
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48C3×#475—91no—
49#4791—31no—
50#4791—91no—
51C2×C32×Q8—21no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C32×Q8C22×C311yescharacteristic
54C2×#634—21no—
55C2×C3×Dic6—61no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C22×C311yescharacteristic
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)C22×C311yescharacteristic
64C4⋊(C3⋊Dic3)C22×C311yescharacteristic
65#3750—21no—
66C4×C3⋊Dic3—21no—
67C4×C3⋊Dic3C22×C311yescharacteristic
68C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×Dic3⋊C4—61no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)—22no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)C2×Q811yescharacteristic
80C4×H(3,𝔽3)#4111yescharacteristic
81(C3⋊Dic3)⋊C3—21no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3SD1612yes—
85C3×C4⋊Q8—91no—
86C3×C4×Q8—91no—
87Q8×Dic3—31no—
88Q8×Dic3—91no—
89#888—31no—
90#888—91no—
91C3×#106—91no—
92C3×#106—91no—
93C3×C4⋊C8—91no—
94C3×#103—31no—
95C3×#103—31no—
96C3×#103—61no—
97C3×#103—91no—
98C3×#103C3×D311yescharacteristic
99#835—31no—
100#835—91no—
101#817—31no—
102#817—91no—
103Dic3⋊C8—31no—
104Dic3⋊C8—91no—
105#808—31no—
106#808—91no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—22no—
110C32×Q8—41no—
111C32×Q8Dic612yescomplemented
112C32×Q8C3×Q812yescomplemented
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C23×C311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3C3×D411yescharacteristic
125C2×C3⋊Dic3C3×D411yescharacteristic
126#634—42no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×Dic6—122no—
132C32×C8—21no—
133C32×C8—21no—
134C32×C8—41no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—61no—
137#475—91noSylow
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)C2×SD1611yescharacteristic
140C2×H(3,𝔽3)#13711yescharacteristic
141C2×C3×Q8—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8—181no—
145C2×C3×Q8C2×C3×D311yescharacteristic
146C2×Dic6—61no—
147C2×Dic6—181no—
148C2×C3×C8—31no—
149C2×C3×C8—31no—
150C2×C3×C8—61no—
151C2×C3×C8—91no—
152C2×C3×C8C2×C3×D311yescharacteristic
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—181no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4C2×C3×D311yescharacteristic
161C3×C42—91no—
162C3×C42—181no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—61no—
166C4⋊Dic3—91no—
167C4⋊Dic3—91no—
168C4⋊Dic3—181no—
169Dic3⋊C4—61no—
170Dic3⋊C4—181no—
171C4×Dic3—31no—
172C4×Dic3—61no—
173C4×Dic3—91no—
174C4×Dic3—181no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—91no—
177C22×C32—21no—
178C22×C32D3×D411yescharacteristic
179C22×C32C2×C3×D411yescharacteristic
180C32×C4—22no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—42no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—81no—
188C32×C4C2×Dic611yescharacteristic
189C32×C4#23711yescharacteristic
190C32×C4C2×C3×Q811yescharacteristic
191C32×C4C3×#4111yescharacteristic
192C3⋊Dic3—21no—
193C3⋊Dic3—41no—
194C3⋊Dic3—41no—
195C3⋊Dic3C3×SD1612yes—
196C3×Dic3—32no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C4⋊Q8—91no—
201C4×Q8—91no—
202#106—91no—
203#106—91no—
204C4⋊C8—91no—
205#103—91no—
206#103(C3⋊D3)⋊C311yescharacteristic
207H(3,𝔽3)#47511yescharacteristic, complemented, Sylow
208C3×Q8—21no—
209C3×Q8—32no—
210C3×Q8—32no—
211C3×Q8—61no—
212C3×Q8—61no—
213C3×Q8—62no—
214C3×Q8—121no—
215C3×Q8—362no—
216C3×Q8C3×Dic612yes—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4C22×C3×D311yescharacteristic
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241Dic6—122no—
242Dic6—362no—
243C3×C8—21no—
244C3×C8—61no—
245C3×C8—61no—
246C3×C8—121no—
247C3×C8—181no—
248C3⋊C8—61no—
249C3⋊C8—181no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32#87811yescharacteristic
254C2×C32D3×SD1611yescharacteristic
255C2×C32#91811yescharacteristic
256C2×C32C3×C22⋊Q811yescharacteristic
257C2×C32C2×C3×SD1611yescharacteristic
258C2×C32C3×#13711yescharacteristic
259C2×Q8—181no—
260C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
261C2×C8—91no—
262C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
263C4⋊C4—91no—
264C4⋊C4—91no—
265C4⋊C4—181no—
266C4⋊C4—181no—
267C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
268C42—91no—
269C42—181no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—61no—
273C22×C3C3×D3×D411yescharacteristic
274C3×C4—22no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—41no—
280C3×C4—62no—
281C3×C4—62no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—92no—
285C3×C4—122no—
286C3×C4—121no—
287C3×C4—121no—
288C3×C4—181no—
289C3×C4—241no—
290C3×C4—361no—
291C3×C4—361no—
292C3×C4C2×C3×Dic611yescharacteristic
293C3×C4C3×#23711yescharacteristic
294Dic3—32no—
295Dic3—61no—
296Dic3—92no—
297Dic3—121no—
298Dic3—121no—
299Dic3—181no—
300Dic3—361no—
301Dic3—361no—
302C32—21no—
303C32#479111yescharacteristic, complemented
304C32C3×#47511yescharacteristic, complemented
305Q8—21no—
306Q8—362no—
307Q8#634⋊C312yescomplemented
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
315C8—21no—
316C8—181no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3C3×#87811yescharacteristic
327C2×C3C3×D3×SD1611yescharacteristic
328C2×C3C3×#91811yescharacteristic
329C22D4×(C3⋊D3)⋊C311yescharacteristic
330C4—22no—
331C4—41no—
332C4—92no—
333C4—181no—
334C4—361no—
335C4—361no—
336C4C2×#634⋊C311yescharacteristic
337C4#3830⋊C311yescharacteristic
338C3—31no—
339C3—31no—
340C3—61no—
341C3C3×#479111yescharacteristic
342C2#64087⋊C311yescharacteristic
343C2SD16×(C3⋊D3)⋊C311yescharacteristic
344C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#475
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
8. #10592490C3semidirect
37. Q8×H(3,𝔽3)Q8semidirect
111. C32×Q8Dic6semidirect
112. C32×Q8C3×Q8semidirect
207. H(3,𝔽3)#475semidirect
303. C32#4791semidirect
304. C32C3×#475semidirect
307. Q8#634⋊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
#64097⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63644⋊C32
#63641⋊C32
#63635⋊C32
#105924903
C3×#47913