Finite Group Databasebeta

← #10592718⋊C3 #13142746

Subgroups — #10592718⋊C3 #13142746

Order 1728.

514 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3754⋊C3C212yesmaximal
3#64078⋊C3C211yesmaximal, characteristic, complemented
4#63637⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#10592718C311yesmaximal, characteristic, complemented
7C3×#5019—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)—21no—
10C2×#634⋊C3C2212yes—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C22⋊C4×H(3,𝔽3)—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14#3754⋊C3—24no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C32×C4×D4—21no—
19C32×C4×D4D311yescharacteristic
20C32×C4×D4C2×C311yescharacteristic
21C2×#3754C2×C312yes—
22#64078C2×C311yescharacteristic, complemented
23C2×C3×#214—32no—
24C3×#869—31no—
25#63637C2×C312yescomplemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#810—32no—
28C3×C4⋊(C3⋊C8)—31no—
29C23×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)—21no—
31D4×H(3,𝔽3)D412yes—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—42no—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×(C3⋊Dic3)⋊C3—22no—
38#634⋊C3—22no—
39#634⋊C3—22no—
40(C3⋊(C3⋊C8))⋊C3—22no—
41C3×#461—91no—
42#5019—31no—
43#5019—91no—
44C2×C32×D4—21no—
45C2×C32×D4C2×D311yescharacteristic
46C2×C32×D4C22×C311yescharacteristic
47C22×C32×C4—22no—
48C22×C32×C4—21no—
49C22×C32×C4—21no—
50C2×#634C22×C312yes—
51C2×C3×Dic6—32no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C22×C311yescharacteristic
55C32×C22⋊C4—22no—
56C32×C22⋊C4—21no—
57C32×C22⋊C4—21no—
58C32×C42—21no—
59C32×C42C2×D311yescharacteristic
60C32×C42C22×C311yescharacteristic
61#3754—24no—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)—21no—
64C2×C3⋊(C3⋊C8)C22×C312yes—
65C3×#214—64no—
66C3×C4⋊Dic3—61no—
67C3×C4⋊Dic3—61no—
68C2×C3×C3⋊C8—32no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)C2×D411yescharacteristic
73C4×H(3,𝔽3)—21no—
74C4×H(3,𝔽3)—41no—
75C4×H(3,𝔽3)SD1612yescomplemented
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)#4111yescharacteristic
78(C3⋊Dic3)⋊C3—42no—
79C2×C3×SD16—92no—
80C3×C4⋊Q8—91no—
81C3×C4×D4—31no—
82C3×C4×D4—31no—
83C3×C4×D4—61no—
84C3×C4×D4C3×D311yescharacteristic
85C2×#214—32no—
86C2×#214—92no—
87#869—31no—
88#869—91no—
89C3×C4⋊C8—91no—
90C3×#103—92no—
91#810—32no—
92#810—92no—
93C4⋊(C3⋊C8)—31no—
94C4⋊(C3⋊C8)—91no—
95C23×C32—22no—
96C23×C32—21no—
97C23×C32—21no—
98C32×D4—22no—
99C32×D4—21no—
100C32×D4—21no—
101C32×D4—22no—
102C32×D4D1212yes—
103C32×D4C3×D412yes—
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—44no—
111C2×C32×C4—42no—
112C2×C32×C4—42no—
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4C3×D411yescharacteristic
116C2×C32×C4C3×D411yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C23×C311yescharacteristic
119C2×C3⋊Dic3—22no—
120#634—22no—
121#634—22no—
122C2×C3×Dic3—62no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125C3⋊(C3⋊C8)—22no—
126C3×C3⋊C8—62no—
127#461—91noSylow
128C2×H(3,𝔽3)—42no—
129C2×H(3,𝔽3)#12111yescharacteristic
130C2×H(3,𝔽3)C2×SD1611yescharacteristic
131C2×H(3,𝔽3)#13711yescharacteristic
132C2×C3×Q8—92no—
133C2×C3×D4—31no—
134C2×C3×D4—31no—
135C2×C3×D4—61no—
136C2×C3×D4C2×C3×D311yescharacteristic
137C22×C3×C4—21no—
138C22×C3×C4—61no—
139C22×C3×C4—61no—
140C22×C3×C4—62no—
141C2×Dic6—32no—
142C2×Dic6—92no—
143C3×SD16—184no—
144C2×C3×C8—92no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—61no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4C2×C3×D311yescharacteristic
151C3×C22⋊C4—21no—
152C3×C22⋊C4—61no—
153C3×C22⋊C4—61no—
154C3×C22⋊C4—62no—
155C3×C42—31no—
156C3×C42—31no—
157C3×C42—61no—
158C3×C42C2×C3×D311yescharacteristic
159#214—64no—
160#214—184no—
161C4⋊Dic3—61no—
162C4⋊Dic3—61no—
163C4⋊Dic3—181no—
164C4⋊Dic3—181no—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—92no—
167C22×C32—24no—
168C22×C32—22no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32C2×#6711yescharacteristic
178C22×C32C2×C3×D411yescharacteristic
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—42no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4#21412yescomplemented
189C32×C4C3×SD1612yescomplemented
190C32×C4C2×D1211yescharacteristic
191C32×C4#23511yescharacteristic
192C32×C4C2×C3×D411yescharacteristic
193C32×C4C3×#4111yescharacteristic
194C3⋊Dic3—42no—
195C3×Dic3—122no—
196C2×SD16—92no—
197C4⋊Q8—91no—
198C4×D4(C3⋊D3)⋊C311yescharacteristic
199C4⋊C8—91no—
200#103—92no—
201H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
202C23×C3—21no—
203C23×C3—61no—
204C23×C3—61no—
205C23×C3—62no—
206C3×Q8—182no—
207C3×Q8—182no—
208C3×D4—21no—
209C3×D4—32no—
210C3×D4—32no—
211C3×D4—61no—
212C3×D4—61no—
213C3×D4—62no—
214C3×D4—62no—
215C3×D4C3×D1212yes—
216C2×C3×C4—21no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—42no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—62no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—122no—
231C2×C3×C4—122no—
232C2×C3×C4—124no—
233C2×C3×C4—182no—
234C2×C3×C4C3×#6711yescharacteristic
235C2×C3×C4C3×#6711yescharacteristic
236C2×C3×C4C22×C3×D311yescharacteristic
237C2×Dic3—62no—
238C2×Dic3—182no—
239Dic6—62no—
240Dic6—62no—
241Dic6—182no—
242Dic6—182no—
243C3×C8—182no—
244C3⋊C8—62no—
245C3⋊C8—182no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—44no—
250C2×C32—42no—
251C2×C32—42no—
252C2×C32#93011yescharacteristic
253C2×C32C2×#21411yescharacteristic
254C2×C32#95111yescharacteristic
255C2×C32C3×#12111yescharacteristic
256C2×C32C2×C3×SD1611yescharacteristic
257C2×C32C3×#13711yescharacteristic
258C2×Q8—92no—
259C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
260C22×C4—21no—
261SD16—184no—
262C2×C8—92no—
263C4⋊C4—181no—
264C4⋊C4—181no—
265C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
266C22⋊C4—21no—
267C42C2×(C3⋊D3)⋊C311yescharacteristic
268C22×C3—22no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—62no—
274C22×C3—62no—
275C22×C3—64no—
276C22×C3—61no—
277C22×C3—121no—
278C22×C3—121no—
279C22×C3—121no—
280C22×C3—121no—
281C22×C3—122no—
282C22×C3—122no—
283C22×C3C2×C3×#6711yescharacteristic
284C3×C4—21no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—32no—
290C3×C4—32no—
291C3×C4—41no—
292C3×C4—61no—
293C3×C4—61no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—62no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—121no—
300C3×C4—122no—
301C3×C4—362no—
302C3×C4C3×#21412yes—
303C3×C4C2×C3×D1211yescharacteristic
304C3×C4C3×#23511yescharacteristic
305Dic3—122no—
306Dic3—362no—
307C32—21no—
308C32#501911yescharacteristic, complemented
309C32C3×#46111yescharacteristic, complemented
310C23—21no—
311Q8—182no—
312Q8—182no—
313D4—21no—
314D4C4⋊((C3⋊D3)⋊C3)12yes—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—182no—
318C2×C4#638⋊C311yescharacteristic
319C2×C4#638⋊C311yescharacteristic
320C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
321C8—182no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—42no—
329C2×C3—61no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—122no—
333C2×C3—122no—
334C2×C3—124no—
335C2×C3C3×#93011yescharacteristic
336C2×C3C2×C3×#21411yescharacteristic
337C2×C3C3×#95111yescharacteristic
338C22—22no—
339C22—41no—
340C22—41no—
341C22C2×#638⋊C311yescharacteristic
342C4—21no—
343C4—41no—
344C4—362no—
345C4#3754⋊C312yescomplemented
346C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
347C4#3828⋊C311yescharacteristic
348C3—31no—
349C3—31no—
350C3—61no—
351C3C3×#501911yescharacteristic
352C2—42no—
353C2#64139⋊C311yescharacteristic
354C2C2×#3754⋊C311yescharacteristic
355C2#64160⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
3. #64078⋊C3C2semidirect
4. #63637⋊C3C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #10592718C3semidirect
22. #64078C2×C3semidirect
25. #63637C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
75. C4×H(3,𝔽3)SD16semidirect
188. C32×C4#214semidirect
189. C32×C4C3×SD16semidirect
201. H(3,𝔽3)#461semidirect
308. C32#5019semidirect
309. C32C3×#461semidirect
345. C4#3754⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
C2×#3754⋊C32
#64078⋊C32
#63637⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927183
C3×#50193