Finite Group Databasebeta

← #10592445⋊C3 #13141674

Subgroups — #10592445⋊C3 #13141674

Order 1728.

578 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 90 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#64361⋊C3C211yesmaximal, characteristic
2#64145⋊C3C211yesmaximal, characteristic, complemented
3#64118⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
5#63652⋊C3C212yesmaximal, complemented
6#10592445C311yesmaximal, characteristic, complemented
7C3×#4746—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—22no—
10#3830⋊C3—22no—
11D4×(C3⋊D3)⋊C3—22no—
12#3828⋊C3C2212yes—
13C2×#634⋊C3C2211yescharacteristic, complemented
14M16×H(3,𝔽3)C2212yescomplemented
15#3757⋊C3—22no—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
17#3744⋊C3—22no—
18#3746⋊C3—22no—
19#64361C2×C311yescharacteristic
20C3×#1004—31no—
21#64145C2×C311yescharacteristic, complemented
22#64118C2×C312yescomplemented
23C3×#936—31no—
24C3×#909—32no—
25C32×#100—21no—
26C32×#100D311yescharacteristic, complemented
27C32×#100C2×C311yescharacteristic, complemented
28#63652C2×C312yescomplemented
29C3×#825—32no—
30C23×H(3,𝔽3)D412yes—
31C22×(C3⋊D3)⋊C3—22no—
32D4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34#638⋊C3—24no—
35#638⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—21no—
38C2×(C3⋊Dic3)⋊C3—21no—
39C4⋊((C3⋊D3)⋊C3)D412yes—
40C4×(C3⋊D3)⋊C3—22no—
41#634⋊C3—22no—
42#634⋊C3D412yes—
43C8×H(3,𝔽3)—22no—
44C3×#455—91no—
45#4746—31no—
46#4746—91no—
47C2×C32×D4—21no—
48C2×C32×D4C2×D311yescharacteristic
49C2×C32×D4C22×C311yescharacteristic
50C2×#638—22no—
51#3830—22no—
52D4×C3⋊D3—22no—
53#3828C22×C312yes—
54C2×#634C22×C311yescharacteristic, complemented
55C2×C3×#67—62no—
56C3×#237—62no—
57C3×D3×D4—62no—
58C3×#235—32no—
59C2×C3×Dic6—31no—
60C32×M16—22no—
61C32×M16C2×D312yescomplemented
62C32×M16C22×C312yescomplemented
63#3757—22no—
64C4×C3⋊Dic3C22×C311yescharacteristic, complemented
65#3746—22no—
66#3744—22no—
67C3×#217—62no—
68C3×C4×Dic3—31no—
69C3×Dic12—62no—
70C3×#204—62no—
71C22×H(3,𝔽3)—22no—
72C22×H(3,𝔽3)—42no—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—82no—
76C4×H(3,𝔽3)C2×D412yes—
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3—41no—
79(C3⋊Dic3)⋊C3—41no—
80C3×H(5,𝔽2)—91no—
81#1004—31no—
82#1004—91no—
83C3×#137—92no—
84C3×#124—91no—
85#936—31no—
86#936—91no—
87#909—32no—
88#909—92no—
89C3×#104—92no—
90C3×#100—31no—
91C3×#100—31no—
92C3×#100—61no—
93C3×#100C3×D311yescharacteristic
94#825—32no—
95#825—92no—
96C23×C32—22no—
97C23×C32D1212yes—
98C23×C32C3×D412yes—
99C22×C3⋊D3—22no—
100C22×C3×D3—62no—
101C32×D4—22no—
102C32×D4—22no—
103C32×D4—42no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C23×C311yescharacteristic
107#638—24no—
108#638—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C4⋊(C3⋊D3)C3×D412yes—
113C4×C3⋊D3—22no—
114#634—22no—
115#634C3×D412yes—
116C3×#67—64no—
117C3×#67—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C3×D12—32no—
122C3×C4×D3—62no—
123C3×Dic6—32no—
124C3×Dic6—62no—
125C32×C8—22no—
126C32×C8—22no—
127C32×C8—42no—
128#455—91noSylow
129C2×H(3,𝔽3)—21no—
130C2×H(3,𝔽3)—42no—
131C2×H(3,𝔽3)#12011yescharacteristic
132(C3⋊D3)⋊C3—42no—
133C3×#41—92no—
134C3×#41—182no—
135C2×C3×Q8—91no—
136C2×C3×D4—31no—
137C2×C3×D4—31no—
138C2×C3×D4—61no—
139C2×C3×D4—182no—
140C2×C3×D4—182no—
141C2×C3×D4C2×C3×D311yescharacteristic
142C2×#67—62no—
143C2×#67—182no—
144#237—62no—
145#237—182no—
146D3×D4—62no—
147D3×D4—182no—
148#235—32no—
149#235—92no—
150C2×Dic6—31no—
151C2×Dic6—91no—
152C3×Q16—182no—
153C3×SD16—182no—
154C3×M16—32no—
155C3×M16—32no—
156C3×M16—62no—
157C3×M16C2×C3×D312yes—
158C3×C22⋊C4—182no—
159C3×C42—91no—
160#217—62no—
161#217—182no—
162C4×Dic3—31no—
163C4×Dic3—91no—
164Dic12—62no—
165Dic12—182no—
166#204—62no—
167#204—182no—
168C22×C32—22no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—82no—
175C22×C32C2×D1211yescharacteristic
176C22×C32C2×C3×D411yescharacteristic
177C2×C3⋊D3—22no—
178C2×C3⋊D3—82no—
179C2×C3×D3—62no—
180C2×C3×D3—242no—
181C32×C4—22no—
182C32×C4D3×D412yes—
183C32×C4C2×C3×D412yes—
184C3⋊Dic3—22no—
185C3⋊Dic3—41no—
186C3⋊Dic3—41no—
187C3×Dic3—62no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190H(5,𝔽2)—91no—
191#137—92no—
192#124—91no—
193#104—92no—
194#100(C3⋊D3)⋊C311yescharacteristic, complemented
195H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
196C23×C3—32no—
197C23×C3—32no—
198C23×C3—62no—
199C23×C3—182no—
200C23×C3C3×D1212yes—
201C22×D3—62no—
202C22×D3—182no—
203C3×Q8—92no—
204C3×Q8—182no—
205C3×D4—22no—
206C3×D4—62no—
207C3×D4—62no—
208C3×D4—92no—
209C3×D4—122no—
210C3×D4—182no—
211C3×D4—184no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—61no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C22×C3×D311yescharacteristic
220#67—64no—
221#67—62no—
222#67—182no—
223#67—184no—
224C2×Dic3—62no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—182no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230D12—32no—
231D12—92no—
232C4×D3—62no—
233C4×D3—182no—
234Dic6—32no—
235Dic6—62no—
236Dic6—92no—
237Dic6—182no—
238C3×C8—22no—
239C3×C8—62no—
240C3×C8—62no—
241C3×C8—122no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—41no—
246C2×C32—42no—
247C2×C32—42no—
248C2×C32—82no—
249C2×C32#88211yescharacteristic
250C2×C32C3×#12011yescharacteristic
251C3⋊D3—42no—
252C3×D3—122no—
253#41—92no—
254#41—182no—
255C2×Q8—91no—
256C2×D4—182no—
257C2×D4—182no—
258C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
259Q16—182no—
260SD16—182no—
261M16C2×(C3⋊D3)⋊C312yescomplemented
262C22⋊C4—182no—
263C42—91no—
264C22×C3—22no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—42no—
268C22×C3—62no—
269C22×C3—62no—
270C22×C3—61no—
271C22×C3—122no—
272C22×C3—122no—
273C22×C3—122no—
274C22×C3—182no—
275C22×C3—242no—
276C22×C3—722no—
277C22×C3C2×C3×D1211yescharacteristic
278C2×D3—62no—
279C2×D3—182no—
280C2×D3—242no—
281C2×D3—722no—
282C3×C4—32no—
283C3×C4—32no—
284C3×C4—62no—
285C3×C4—182no—
286C3×C4—361no—
287C3×C4—361no—
288C3×C4C3×D3×D412yes—
289Dic3—62no—
290Dic3—121no—
291Dic3—121no—
292Dic3—182no—
293Dic3—361no—
294Dic3—361no—
295C32—21no—
296C32#474611yescharacteristic, complemented
297C32C3×#45511yescharacteristic, complemented
298C23—182no—
299C23C4⋊((C3⋊D3)⋊C3)12yes—
300Q8—92no—
301Q8—182no—
302D4—22no—
303D4—92no—
304D4—184no—
305D4—182no—
306C2×C4—182no—
307C2×C4—182no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
311C8—22no—
312C2×C3—21no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—42no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—121no—
320C2×C3—122no—
321C2×C3—122no—
322C2×C3—242no—
323C2×C3—362no—
324C2×C3C3×#88211yescharacteristic
325D3—122no—
326D3—362no—
327C22—22no—
328C22—42no—
329C22—182no—
330C22—722no—
331C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
332C4—182no—
333C4—361no—
334C4—361no—
335C4D4×(C3⋊D3)⋊C312yes—
336C3—31no—
337C3—31no—
338C3—61no—
339C3C3×#474611yescharacteristic
340C2—21no—
341C2—42no—
342C2—362no—
343C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#100C2
socleC2×C3C3×#882
prime radical#64361⋊C3C2
hypercenter#100(C3⋊D3)⋊C3
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. #64145⋊C3C2semidirect
3. #64118⋊C3C2semidirect
4. H(3,𝔽3)×#100C2semidirect
5. #63652⋊C3C2semidirect
6. #10592445C3semidirect
13. C2×#634⋊C3C22semidirect
14. M16×H(3,𝔽3)C22semidirect
16. C4×(C3⋊Dic3)⋊C3C22semidirect
21. #64145C2×C3semidirect
22. #64118C2×C3semidirect
26. C32×#100D3semidirect
27. C32×#100C2×C3semidirect
28. #63652C2×C3semidirect
54. C2×#634C22×C3semidirect
61. C32×M16C2×D3semidirect
62. C32×M16C22×C3semidirect
64. C4×C3⋊Dic3C22×C3semidirect
194. #100(C3⋊D3)⋊C3semidirect
195. H(3,𝔽3)#455semidirect
261. M16C2×(C3⋊D3)⋊C3semidirect
296. C32#4746semidirect
297. C32C3×#455semidirect

Maximal subgroups

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

subgroupindex
#64361⋊C32
#64145⋊C32
#64118⋊C32
H(3,𝔽3)×#1002
#63652⋊C32
#105924453
C3×#47463