Finite Group Databasebeta

← #10592587⋊C3 #13142229

Subgroups — #10592587⋊C3 #13142229

Order 1728.

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

normal subgroups only

This group has 372 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#64106⋊C3C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#63635⋊C3C211yesmaximal, characteristic, complemented
8#10592587C311yesmaximal, characteristic, complemented
9C3×#4888—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3—21no—
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64106C2×C311yescharacteristic
22#64101C2×C311yescharacteristic
23C3×#897—31no—
24C3×#892—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic, complemented
27C32×#107C2×C311yescharacteristic, complemented
28#63647C2×C311yescharacteristic
29#63643C2×C311yescharacteristic, complemented
30#63637C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C3×(C2×D3)⋊C8—31no—
33C3×#816—31no—
34C3×#810—31no—
35C3×#808—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#485—91no—
50#4888—31no—
51#4888—91no—
52C2×C4×C3⋊D3C22×C311yescharacteristic
53C2×#634C22×C311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×Dic6—31no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64C32×C4⋊C4C22×C311yescharacteristic
65#3752—21no—
66#3752—21no—
67C4⋊(C3⋊Dic3)—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C4×C3⋊Dic3—21no—
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C3×(C2×D3)⋊C4—61no—
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—61no—
75C3×C4×Dic3—61no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—21no—
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88C3×C22⋊Q8—91no—
89C3×#117—91no—
90#897—31no—
91#897—91no—
92#892—31no—
93#892—91no—
94C3×#107—31no—
95C3×#107—31no—
96C3×#107—61no—
97C3×#107C3×D311yescharacteristic
98C3×#106—91no—
99C3×#103—91no—
100C3×#103—91no—
101C3×C22⋊C8—91no—
102(C2×D3)⋊C8—31no—
103(C2×D3)⋊C8—91no—
104#816—31no—
105#816—91no—
106#810—31no—
107#810—91no—
108#808—31no—
109#808—91no—
110C22×C3⋊D3C3×D411yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3C3×D411yescharacteristic
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126#634—21no—
127#634—21no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C32×C8—22no—
136C32×C8—21no—
137C32×C8—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140#485—91noSylow
141C2×H(3,𝔽3)#12311yescharacteristic
142C2×H(3,𝔽3)#13511yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144(C3⋊D3)⋊C3—41no—
145C2×C3×Q8—91no—
146C22×C3×C4—91no—
147C2×C4×D3—31no—
148C2×C4×D3—91no—
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—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—61no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—181no—
164C3×C4⋊C4C2×C3×D311yescharacteristic
165C3×C4⋊C4C2×C3×D311yescharacteristic
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168C3×C42—181no—
169(C2×D3)⋊C4—61no—
170(C2×D3)⋊C4—61no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—61no—
175C4⋊Dic3—91no—
176C4⋊Dic3—181no—
177C4×Dic3—61no—
178C4×Dic3—181no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—91no—
181C22×C32—21no—
182C22×C32D3×D411yescharacteristic
183C22×C32C2×C3×D411yescharacteristic
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3×D3—61no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—81no—
197C32×C4—81no—
198C32×C4#23511yescharacteristic
199C32×C4#23911yescharacteristic
200C32×C4C3×#4111yescharacteristic
201C32×C4C3×#4111yescharacteristic
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3×Dic3—61no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C22⋊Q8—91no—
209#117—91no—
210#107(C3⋊D3)⋊C311yescharacteristic, complemented
211#106—91no—
212#103—91no—
213#103—91no—
214C22⋊C8—91no—
215H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
216C23×C3—91no—
217C22×D3—31no—
218C22×D3—91no—
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—121no—
232C2×C3×C4—121no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4C22×C3×D311yescharacteristic
238C2×Dic3—31no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—91no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C4×D3—61no—
245C4×D3—61no—
246C4×D3—181no—
247C4×D3—181no—
248Dic6—61no—
249Dic6—61no—
250Dic6—181no—
251Dic6—181no—
252C3×C8—21no—
253C3×C8—61no—
254C3×C8—61no—
255C3×C8—62no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32#89411yescharacteristic
263C2×C32#91211yescharacteristic
264C2×C32#91811yescharacteristic
265C2×C32C3×#12311yescharacteristic
266C2×C32C3×#13511yescharacteristic
267C2×C32C3×#13711yescharacteristic
268C3⋊D3—41no—
269C3×D3—121no—
270C2×Q8—91no—
271C22×C4—91no—
272C2×C8—91no—
273C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
274C4⋊C4—91no—
275C4⋊C4—181no—
276C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
277C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
278C22⋊C4—181no—
279C22⋊C4—181no—
280C42—181no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—61no—
284C22×C3—181no—
285C22×C3—181no—
286C22×C3—181no—
287C22×C3C3×D3×D411yescharacteristic
288C2×D3—61no—
289C2×D3—61no—
290C2×D3—61no—
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—181no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—121no—
303C3×C4—121no—
304C3×C4—121no—
305C3×C4—121no—
306C3×C4—181no—
307C3×C4—241no—
308C3×C4—241no—
309C3×C4—361no—
310C3×C4—361no—
311C3×C4C3×#23511yescharacteristic
312C3×C4C3×#23911yescharacteristic
313Dic3—61no—
314Dic3—121no—
315Dic3—121no—
316Dic3—181no—
317Dic3—361no—
318Dic3—361no—
319C32—21no—
320C32#488811yescharacteristic, complemented
321C32C3×#48511yescharacteristic, complemented
322C23—91no—
323Q8—181no—
324Q8—181no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—91no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
333C8—21no—
334C8—181no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—361no—
345C2×C3C3×#89411yescharacteristic
346C2×C3C3×#91211yescharacteristic
347C2×C3C3×#91811yescharacteristic
348D3—121no—
349D3—361no—
350C22—181no—
351C22—181no—
352C22—181no—
353C22D4×(C3⋊D3)⋊C311yescharacteristic
354C4—41no—
355C4—41no—
356C4—181no—
357C4—361no—
358C4—361no—
359C4#3828⋊C311yescharacteristic
360C4#8872211yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3C3×#488811yescharacteristic
365C2—361no—
366C2#64103⋊C311yescharacteristic
367C2#64121⋊C311yescharacteristic
368C2#64127⋊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×(C3⋊D3)⋊C3D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#485
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
3. H(3,𝔽3)×#107C2semidirect
5. #63643⋊C3C2semidirect
6. #63637⋊C3C2semidirect
7. #63635⋊C3C2semidirect
8. #10592587C3semidirect
26. C32×#107D3semidirect
27. C32×#107C2×C3semidirect
29. #63643C2×C3semidirect
30. #63637C2×C3semidirect
31. #63635C2×C3semidirect
210. #107(C3⋊D3)⋊C3semidirect
215. H(3,𝔽3)#485semidirect
320. C32#4888semidirect
321. C32C3×#485semidirect

Maximal subgroups

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

subgroupindex
#64106⋊C32
#64101⋊C32
H(3,𝔽3)×#1072
#63647⋊C32
#63643⋊C32
#63637⋊C32
#63635⋊C32
#105925873
C3×#48883