Finite Group Databasebeta

← #10592574⋊C3 #13142215

Subgroups — #10592574⋊C3 #13142215

Order 1728.

372 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 94 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#64097⋊C3C211yesmaximal, characteristic
2#64096⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#107C211yesmaximal, characteristic
4#63643⋊C3C211yesmaximal, characteristic
5#63641⋊C3C211yesmaximal, characteristic
6#63637⋊C3C211yesmaximal, characteristic
7#63634⋊C3C211yesmaximal, characteristic
8#10592574C311yesmaximal, characteristic, complemented
9C3×#4875—31nomaximal
10C2×#634⋊C3—21no—
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64097C2×C311yescharacteristic
22#64096C2×C311yescharacteristic
23C3×#888—31no—
24C3×#887—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic
27C32×#107C2×C311yescharacteristic
28#63643C2×C311yescharacteristic
29#63641C2×C311yescharacteristic
30#63637C2×C311yescharacteristic
31#63634C2×C311yescharacteristic
32C3×#816—31no—
33C3×Dic3⋊C8—31no—
34C3×#810—31no—
35C3×#807—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43#634⋊C3—21no—
44#634⋊C3—42no—
45#634⋊C3Q812yescomplemented
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48C3×#477—91no—
49#4875—31no—
50#4875—91no—
51C2×#634—21no—
52C2×#634C22×C311yescharacteristic
53C2×C3×Dic6—31no—
54C2×C3×Dic6—61no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C22×C311yescharacteristic
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C22×C311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64C4⋊(C3⋊Dic3)C22×C311yescharacteristic
65#3750—21no—
66#3750—21no—
67C4×C3⋊Dic3—21no—
68C4×C3⋊Dic3C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×C4⋊Dic3—31no—
71C3×Dic3⋊C4—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)—41no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)C2×Q811yescharacteristic
80C4×H(3,𝔽3)#4111yescharacteristic
81(C3⋊Dic3)⋊C3—22no—
82(C3⋊Dic3)⋊C3—21no—
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3Q1612yes—
85C3×C4⋊Q8—91no—
86C3×C4×Q8—91no—
87#888—31no—
88#888—91no—
89#887—31no—
90#887—91no—
91C3×#107—31no—
92C3×#107—31no—
93C3×#107—61no—
94C3×#107—91no—
95C3×#107C3×D311yescharacteristic
96C3×C4⋊C8—91no—
97C3×#103—91no—
98C3×#103—91no—
99#816—31no—
100#816—91no—
101Dic3⋊C8—31no—
102Dic3⋊C8—91no—
103#810—31no—
104#810—91no—
105#807—31no—
106#807—91no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—41no—
113C2×C32×C4—41no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C23×C311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3C3×D411yescharacteristic
119C2×C3⋊Dic3C3×D411yescharacteristic
120#634—21no—
121#634—42no—
122#634C3×Q812yescomplemented
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C3×Dic6—32no—
128C3×Dic6—61no—
129C3×Dic6—122no—
130C32×C8—22no—
131C32×C8—21no—
132C32×C8—21no—
133C3⋊(C3⋊C8)—21no—
134C3×C3⋊C8—61no—
135#477—91noSylow
136C2×H(3,𝔽3)C22⋊Q811yescharacteristic
137C2×H(3,𝔽3)C2×Q1611yescharacteristic
138C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
139C2×C3×Q8—91no—
140C2×C3×Q8—181no—
141C2×Dic6—31no—
142C2×Dic6—61no—
143C2×Dic6—91no—
144C2×Dic6—181no—
145C2×C3×C8—31no—
146C2×C3×C8—31no—
147C2×C3×C8—61no—
148C2×C3×C8—91no—
149C2×C3×C8C2×C3×D311yescharacteristic
150C3×C4⋊C4—31no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—61no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4—181no—
159C3×C4⋊C4C2×C3×D311yescharacteristic
160C3×C4⋊C4C2×C3×D311yescharacteristic
161C3×C42—91no—
162C3×C42—181no—
163C4⋊Dic3—31no—
164C4⋊Dic3—91no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169C4×Dic3—31no—
170C4×Dic3—61no—
171C4×Dic3—91no—
172C4×Dic3—181no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—91no—
175C22×C32—21no—
176C22×C32D3×D411yescharacteristic
177C22×C32C2×C3×D411yescharacteristic
178C32×C4—21no—
179C32×C4—21no—
180C32×C4—41no—
181C32×C4—41no—
182C32×C4—41no—
183C32×C4—41no—
184C32×C4—81no—
185C32×C4—81no—
186C32×C4#23511yescharacteristic
187C32×C4D3×Q811yescharacteristic
188C32×C4C2×C3×Q811yescharacteristic
189C32×C4C3×#4111yescharacteristic
190C3⋊Dic3—22no—
191C3⋊Dic3—21no—
192C3⋊Dic3—41no—
193C3⋊Dic3C3×Q1612yes—
194C3×Dic3—32no—
195C3×Dic3—62no—
196C3×Dic3—61no—
197C3×Dic3—121no—
198C4⋊Q8—91no—
199C4×Q8—91no—
200#107—91no—
201#107(C3⋊D3)⋊C311yescharacteristic
202C4⋊C8—91no—
203#103—91no—
204#103—91no—
205H(3,𝔽3)#47711yescharacteristic, complemented, Sylow
206C3×Q8—92no—
207C3×Q8—181no—
208C3×Q8—362no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—91no—
219C2×C3×C4—91no—
220C2×C3×C4—121no—
221C2×C3×C4—121no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4C22×C3×D311yescharacteristic
225C2×Dic3—31no—
226C2×Dic3—31no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—91no—
230C2×Dic3—91no—
231C2×Dic3—181no—
232C2×Dic3—181no—
233Dic6—32no—
234Dic6—61no—
235Dic6—92no—
236Dic6—122no—
237Dic6—181no—
238Dic6—362no—
239C3×C8—21no—
240C3×C8—61no—
241C3×C8—61no—
242C3×C8—62no—
243C3×C8—181no—
244C3⋊C8—61no—
245C3⋊C8—181no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32#89611yescharacteristic
250C2×C32#91111yescharacteristic
251C2×C32D3×Q1611yescharacteristic
252C2×C32C3×C22⋊Q811yescharacteristic
253C2×C32C2×C3×Q1611yescharacteristic
254C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
255C2×Q8—91no—
256C2×Q8—181no—
257C2×C8—91no—
258C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
259C4⋊C4—91no—
260C4⋊C4—181no—
261C4⋊C4—181no—
262C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
263C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
264C42—91no—
265C42—181no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—61no—
269C22×C3C3×D3×D411yescharacteristic
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—92no—
279C3×C4—121no—
280C3×C4—121no—
281C3×C4—121no—
282C3×C4—121no—
283C3×C4—181no—
284C3×C4—182no—
285C3×C4—241no—
286C3×C4—241no—
287C3×C4—361no—
288C3×C4C3×#23511yescharacteristic
289C3×C4C3×D3×Q811yescharacteristic
290Dic3—32no—
291Dic3—62no—
292Dic3—61no—
293Dic3—92no—
294Dic3—121no—
295Dic3—182no—
296Dic3—181no—
297Dic3—361no—
298C32—21no—
299C32#487511yescharacteristic, complemented
300C32C3×#47711yescharacteristic, complemented
301Q8—92no—
302Q8—181no—
303Q8—362no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
311C8—21no—
312C8—181no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—61no—
320C2×C3—61no—
321C2×C3—61no—
322C2×C3C3×#89611yescharacteristic
323C2×C3C3×#91111yescharacteristic
324C2×C3C3×D3×Q1611yescharacteristic
325C22D4×(C3⋊D3)⋊C311yescharacteristic
326C4—41no—
327C4—41no—
328C4—92no—
329C4—181no—
330C4—182no—
331C4—361no—
332C4#3828⋊C311yescharacteristic
333C4Q8×(C3⋊D3)⋊C311yescharacteristic
334C3—31no—
335C3—31no—
336C3—61no—
337C3C3×#487511yescharacteristic
338C2#64105⋊C311yescharacteristic
339C2#64120⋊C311yescharacteristic
340C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#477
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
8. #10592574C3semidirect
45. #634⋊C3Q8semidirect
122. #634C3×Q8semidirect
205. H(3,𝔽3)#477semidirect
299. C32#4875semidirect
300. C32C3×#477semidirect

Maximal subgroups

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

subgroupindex
#64097⋊C32
#64096⋊C32
H(3,𝔽3)×#1072
#63643⋊C32
#63641⋊C32
#63637⋊C32
#63634⋊C32
#105925743
C3×#48753