Finite Group Databasebeta

← #10592695⋊C3 #13142693

Subgroups — #10592695⋊C3 #13142693

Order 1728.

412 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#117C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#63637⋊C3C211yesmaximal, characteristic, complemented
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592695C311yesmaximal, characteristic, complemented
9C3×#4996—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)—21no—
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C32×#117—21no—
22C32×#117D311yescharacteristic
23C32×#117C2×C311yescharacteristic
24#64133C2×C311yescharacteristic
25C3×#924—31no—
26#63657C2×C311yescharacteristic
27#63637C2×C311yescharacteristic, complemented
28#63637C2×C311yescharacteristic, complemented
29#63635C2×C311yescharacteristic, complemented
30#63634C2×C311yescharacteristic, complemented
31C3×#830—31no—
32C3×#810—31no—
33C3×#810—31no—
34C3×#808—31no—
35C3×#807—31no—
36C23×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)D411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#485—91no—
50#4996—31no—
51#4996—91no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C2×#634C22×C311yescharacteristic
56C2×C3×Dic6—31no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C22×C311yescharacteristic
62C32×C4⋊C4C22×C311yescharacteristic
63C32×C22⋊C4—22no—
64C32×C22⋊C4—21no—
65C32×C22⋊C4—21no—
66C32×C42—22no—
67C32×C42—21no—
68C32×C42—21no—
69#3757—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71#3750—21no—
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×#217—61no—
75C3×C4⋊Dic3—31no—
76C3×Dic3⋊C4—61no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90C3×C22⋊Q8—91no—
91C3×#117—31no—
92C3×#117—31no—
93C3×#117—61no—
94C3×#117C3×D311yescharacteristic
95#924—31no—
96#924—91no—
97C3×#107—91no—
98C3×#106—91no—
99C3×#103—91no—
100C3×#103—91no—
101C3×C22⋊C8—91no—
102#830—31no—
103#830—91no—
104#810—31no—
105#810—31no—
106#810—91no—
107#810—91no—
108#808—31no—
109#808—91no—
110#807—31no—
111#807—91no—
112C23×C32—21no—
113C23×C32#6711yescharacteristic
114C23×C32C3×D411yescharacteristic
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4C3×D411yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C23×C311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135#634—21no—
136#634—21no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C3×Dic6—61no—
140C3×Dic6—61no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—21no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145#485—91noSylow
146C2×H(3,𝔽3)—41no—
147C2×H(3,𝔽3)#12311yescharacteristic
148C2×H(3,𝔽3)#13511yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150C2×C3×Q8—91no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—61no—
154C22×C3×C4C2×C3×D311yescharacteristic
155C2×Dic6—31no—
156C2×Dic6—91no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4C2×C3×D311yescharacteristic
168C3×C4⋊C4C2×C3×D311yescharacteristic
169C3×C22⋊C4—21no—
170C3×C22⋊C4—61no—
171C3×C22⋊C4—61no—
172C3×C22⋊C4—62no—
173C3×C22⋊C4—181no—
174C3×C42—21no—
175C3×C42—61no—
176C3×C42—61no—
177C3×C42—62no—
178#217—61no—
179#217—181no—
180C4⋊Dic3—31no—
181C4⋊Dic3—91no—
182Dic3⋊C4—61no—
183Dic3⋊C4—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—91no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32C2×#6711yescharacteristic
199C22×C32C2×C3×D411yescharacteristic
200C32×C4—22no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—42no—
206C32×C4—42no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—41no—
210C32×C4—41no—
211C32×C4#23511yescharacteristic
212C32×C4#23511yescharacteristic
213C32×C4C3×#4111yescharacteristic
214C32×C4C3×#4111yescharacteristic
215C3⋊Dic3—41no—
216C3⋊Dic3—41no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C22⋊Q8—91no—
220#117(C3⋊D3)⋊C311yescharacteristic
221#107—91no—
222#106—91no—
223#103—91no—
224#103—91no—
225C22⋊C8—91no—
226H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—61no—
230C23×C3C3×#6711yescharacteristic
231C3×Q8—181no—
232C3×Q8—181no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4C3×#6711yescharacteristic
258C2×C3×C4C22×C3×D311yescharacteristic
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263Dic6—61no—
264Dic6—61no—
265Dic6—181no—
266Dic6—181no—
267C3×C8—181no—
268C3×C8—181no—
269C3⋊C8—61no—
270C3⋊C8—61no—
271C3⋊C8—181no—
272C3⋊C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—42no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32#92911yescharacteristic
280C2×C32#95011yescharacteristic
281C2×C32#95111yescharacteristic
282C2×C32C3×#12311yescharacteristic
283C2×C32C3×#13511yescharacteristic
284C2×C32C3×#13711yescharacteristic
285C2×Q8—91no—
286C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
287C2×C8—91no—
288C2×C8—91no—
289C4⋊C4—91no—
290C4⋊C4—181no—
291C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
292C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—181no—
295C42—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3C2×C3×#6711yescharacteristic
312C3×C4—21no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—62no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—121no—
325C3×C4—121no—
326C3×C4—121no—
327C3×C4—121no—
328C3×C4—122no—
329C3×C4—122no—
330C3×C4—361no—
331C3×C4—361no—
332C3×C4C3×#23511yescharacteristic
333C3×C4C3×#23511yescharacteristic
334Dic3—121no—
335Dic3—121no—
336Dic3—361no—
337Dic3—361no—
338C32—21no—
339C32#499611yescharacteristic, complemented
340C32C3×#48511yescharacteristic, complemented
341C23#638⋊C311yescharacteristic
342Q8—181no—
343Q8—181no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4#638⋊C311yescharacteristic
351C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
352C8—181no—
353C8—181no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—41no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—61no—
364C2×C3—121no—
365C2×C3—121no—
366C2×C3—122no—
367C2×C3C3×#92911yescharacteristic
368C2×C3C3×#95011yescharacteristic
369C2×C3C3×#95111yescharacteristic
370C22—21no—
371C22—21no—
372C22—21no—
373C22C2×#638⋊C311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—361no—
378C4—361no—
379C4#3828⋊C311yescharacteristic
380C4#3828⋊C311yescharacteristic
381C3—31no—
382C3—31no—
383C3—61no—
384C3C3×#499611yescharacteristic
385C2—41no—
386C2#64138⋊C311yescharacteristic
387C2#64160⋊C311yescharacteristic
388C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#117C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #63637⋊C3C2semidirect
5. #63637⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592695C3semidirect
27. #63637C2×C3semidirect
28. #63637C2×C3semidirect
29. #63635C2×C3semidirect
30. #63634C2×C3semidirect
226. H(3,𝔽3)#485semidirect
339. C32#4996semidirect
340. C32C3×#485semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
#64133⋊C32
#63657⋊C32
#63637⋊C32
#63635⋊C32
#63634⋊C32
#105926953
C3×#49963