Finite Group Databasebeta

← #10592496⋊C3 #13141908

Subgroups — #10592496⋊C3 #13141908

Order 1728.

357 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 86 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#64153⋊C3C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic
6#63642⋊C3C211yesmaximal, characteristic
7#63637⋊C3C211yesmaximal, characteristic
8#10592496C311yesmaximal, characteristic, complemented
9C3×#4797—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17#3750⋊C3—21no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64153C2×C311yescharacteristic
22#64099C2×C311yescharacteristic
23C3×#944—31no—
24C3×#890—31no—
25C32×#103—21no—
26C32×#103D311yescharacteristic
27C32×#103C2×C311yescharacteristic
28#63662C2×C311yescharacteristic
29#63643C2×C311yescharacteristic
30#63642C2×C311yescharacteristic
31#63637C2×C311yescharacteristic
32C3×#835—31no—
33C3×#816—31no—
34C3×#815—31no—
35C3×#810—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)—21no—
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—21no—
46#634⋊C3—21no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#491—91no—
50#4797—31no—
51#4797—91no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C32×Q8C22×C311yescharacteristic
55C2×#634C22×C311yescharacteristic
56C2×C3×Dic6—31no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C22×C311yescharacteristic
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C22×C311yescharacteristic
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)C22×C311yescharacteristic
65#3750—21no—
66#3750—21no—
67#3750—21no—
68C4×C3⋊Dic3C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74C3×Dic3⋊C4—61no—
75C3×C4×Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)#4111yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82(C3⋊Dic3)⋊C3—21no—
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86C3×C4⋊Q8—91no—
87C3×#125—91no—
88#944—31no—
89#944—91no—
90#890—31no—
91#890—91no—
92C3×#103—31no—
93C3×#103—31no—
94C3×#103—61no—
95C3×#103—91no—
96C3×#103—91no—
97C3×#103—91no—
98C3×#103C3×D311yescharacteristic
99C3×#97—91no—
100#835—31no—
101#835—91no—
102#816—31no—
103#816—91no—
104#815—31no—
105#815—91no—
106#810—31no—
107#810—91no—
108C32×Q8—22no—
109C32×Q8—21no—
110C32×Q8—21no—
111C32×Q8—21no—
112C32×Q8—21no—
113C32×Q8—41no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C23×C311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3C3×D411yescharacteristic
126C2×C3⋊Dic3C3×D411yescharacteristic
127#634—21no—
128#634—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C32×C8—21no—
136C32×C8—21no—
137C32×C8—41no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140#491—91noSylow
141C2×H(3,𝔽3)#12411yescharacteristic
142C2×H(3,𝔽3)#13711yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144C2×C3×Q8—31no—
145C2×C3×Q8—31no—
146C2×C3×Q8—61no—
147C2×C3×Q8—91no—
148C2×C3×Q8C2×C3×D311yescharacteristic
149C2×Dic6—31no—
150C2×Dic6—91no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—61no—
154C2×C3×C8—91no—
155C2×C3×C8C2×C3×D311yescharacteristic
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4—181no—
163C3×C4⋊C4—181no—
164C3×C4⋊C4C2×C3×D311yescharacteristic
165C3×C42—91no—
166C4⋊Dic3—31no—
167C4⋊Dic3—61no—
168C4⋊Dic3—91no—
169C4⋊Dic3—181no—
170Dic3⋊C4—61no—
171Dic3⋊C4—61no—
172Dic3⋊C4—61no—
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—91no—
180C22×C32—21no—
181C22×C32D3×D411yescharacteristic
182C22×C32C2×C3×D411yescharacteristic
183C32×C4—21no—
184C32×C4—21no—
185C32×C4—42no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—81no—
191C32×C4#23511yescharacteristic
192C32×C4#23711yescharacteristic
193C32×C4C3×#4111yescharacteristic
194C32×C4C3×#4111yescharacteristic
195C3⋊Dic3—21no—
196C3⋊Dic3—21no—
197C3⋊Dic3—41no—
198C3⋊Dic3—41no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C4⋊Q8—91no—
204#125—91no—
205#103—91no—
206#103—91no—
207#103—91no—
208#103(C3⋊D3)⋊C311yescharacteristic
209#97—91no—
210H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
211C3×Q8—21no—
212C3×Q8—21no—
213C3×Q8—61no—
214C3×Q8—61no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—62no—
218C3×Q8—121no—
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—121no—
233C2×C3×C4—121no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4C22×C3×D311yescharacteristic
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245Dic6—61no—
246Dic6—61no—
247Dic6—181no—
248Dic6—181no—
249C3×C8—21no—
250C3×C8—61no—
251C3×C8—61no—
252C3×C8—121no—
253C3×C8—181no—
254C3⋊C8—61no—
255C3⋊C8—181no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32#88511yescharacteristic
260C2×C32#91511yescharacteristic
261C2×C32#91811yescharacteristic
262C2×C32C3×#12411yescharacteristic
263C2×C32C3×#13711yescharacteristic
264C2×C32C3×#13711yescharacteristic
265C2×Q8—91no—
266C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
267C2×C8—91no—
268C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
269C4⋊C4—91no—
270C4⋊C4—181no—
271C4⋊C4—181no—
272C4⋊C4—181no—
273C4⋊C4—181no—
274C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
275C42—91no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—61no—
279C22×C3C3×D3×D411yescharacteristic
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—61no—
287C3×C4—61no—
288C3×C4—121no—
289C3×C4—121no—
290C3×C4—121no—
291C3×C4—121no—
292C3×C4—122no—
293C3×C4—181no—
294C3×C4—181no—
295C3×C4—241no—
296C3×C4—361no—
297C3×C4—361no—
298C3×C4C3×#23511yescharacteristic
299C3×C4C3×#23711yescharacteristic
300Dic3—61no—
301Dic3—61no—
302Dic3—121no—
303Dic3—121no—
304Dic3—181no—
305Dic3—181no—
306Dic3—361no—
307Dic3—361no—
308C32—21no—
309C32#479711yescharacteristic, complemented
310C32C3×#49111yescharacteristic, complemented
311Q8—21no—
312Q8—21no—
313Q8—181no—
314Q8—181no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—91no—
318C2×C4—91no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
322C8—21no—
323C8—181no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3C3×#88511yescharacteristic
334C2×C3C3×#91511yescharacteristic
335C2×C3C3×#91811yescharacteristic
336C22D4×(C3⋊D3)⋊C311yescharacteristic
337C4—41no—
338C4—41no—
339C4—181no—
340C4—181no—
341C4—361no—
342C4—361no—
343C4#3828⋊C311yescharacteristic
344C4#3830⋊C311yescharacteristic
345C3—31no—
346C3—31no—
347C3—61no—
348C3C3×#479711yescharacteristic
349C2#64094⋊C311yescharacteristic
350C2#64124⋊C311yescharacteristic
351C2#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×#491
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. #10592496C3semidirect
210. H(3,𝔽3)#491semidirect
309. C32#4797semidirect
310. C32C3×#491semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
#64099⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63643⋊C32
#63642⋊C32
#63637⋊C32
#105924963
C3×#47973