Finite Group Databasebeta

← #10592546⋊C3 #13142066

Subgroups — #10592546⋊C3 #13142066

Order 1728.

395 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 96 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63648⋊C3C211yesmaximal, characteristic
6#63645⋊C3C211yesmaximal, characteristic, complemented
7#63644⋊C3C211yesmaximal, characteristic, complemented
8#10592546C311yesmaximal, characteristic, complemented
9C3×#4847—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20#3750⋊C3—21no—
21C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
22#64079C2×C311yescharacteristic, complemented
23C3×C4×D12—31no—
24C3×#870—31no—
25C32×C4⋊C8—21no—
26C32×C4⋊C8D311yescharacteristic, complemented
27C32×C4⋊C8C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic
29#63648C2×C311yescharacteristic
30#63645C2×C311yescharacteristic, complemented
31#63644C2×C311yescharacteristic, complemented
32C3×#821—31no—
33C3×#821—31no—
34C3×#818—31no—
35C3×#817—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yes—
45C4×(C3⋊D3)⋊C3—42no—
46C8×H(3,𝔽3)—21no—
47C8×H(3,𝔽3)—21no—
48C3×#478—91no—
49#4847—31no—
50#4847—91no—
51C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
52C2×C4×C3⋊D3—21no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—61no—
55C2×C32×C8—21no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C22×C311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C32×C42—21no—
62C32×C42C2×D311yescharacteristic
63C32×C42C22×C311yescharacteristic
64#3752—21no—
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68#3750—21no—
69#3750—21no—
70C3×(C2×D3)⋊C4—61no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×(C3⋊D3)⋊C3—22no—
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C4×H(3,𝔽3)—21no—
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)C2×Q811yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87C3×#125—91no—
88C3×C4×D4—91no—
89C4×D12—31no—
90C4×D12—91no—
91#870—31no—
92#870—91no—
93C3×#107—91no—
94C3×#106—91no—
95C3×C4⋊C8—31no—
96C3×C4⋊C8—31no—
97C3×C4⋊C8—61no—
98C3×C4⋊C8C3×D311yescharacteristic
99C3×#102—91no—
100C3×#102—91no—
101#821—31no—
102#821—31no—
103#821—91no—
104#821—91no—
105#818—31no—
106#818—91no—
107#817—31no—
108#817—91no—
109C22×C3⋊D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4D1211yescharacteristic
115C2×C32×C4D1211yescharacteristic
116C2×C32×C4C3×D411yescharacteristic
117C2×C32×C4C3×D411yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C4⋊(C3⋊D3)—21no—
124C4⋊(C3⋊D3)C3×Q812yes—
125C4×C3⋊D3—42no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×D12—32no—
130C3×D12—61no—
131C3×C4×D3—122no—
132C32×C8—21no—
133C32×C8—21no—
134C32×C8—21no—
135C32×C8—21no—
136C32×C8—41no—
137C32×C8—41no—
138#478—91noSylow
139C2×H(3,𝔽3)C22⋊Q811yescharacteristic
140C2×H(3,𝔽3)#13511yescharacteristic
141C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
142(C3⋊D3)⋊C3—42no—
143C2×C3×D4—91no—
144C22×C3×C4—181no—
145C2×D12—31no—
146C2×D12—91no—
147C2×C4×D3—61no—
148C2×C4×D3—181no—
149C2×C3×C8—31no—
150C2×C3×C8—31no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—61no—
154C2×C3×C8—61no—
155C2×C3×C8C2×C3×D311yescharacteristic
156C2×C3×C8C2×C3×D311yescharacteristic
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4—181no—
162C3×C22⋊C4—181no—
163C3×C42—31no—
164C3×C42—31no—
165C3×C42—61no—
166C3×C42C2×C3×D311yescharacteristic
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174C4⋊Dic3—91no—
175Dic3⋊C4—61no—
176Dic3⋊C4—61no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C22×C32—21no—
180C22×C32C2×D1211yescharacteristic
181C22×C32C2×C3×D411yescharacteristic
182C2×C3⋊D3—22no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—62no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C32×C4—22no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4#23711yescharacteristic
197C32×C4D3×Q811yescharacteristic
198C32×C4C2×C3×Q811yescharacteristic
199C32×C4C3×#4111yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3⋊Dic3—41no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206#125—91no—
207C4×D4—91no—
208#107—91no—
209#106—91no—
210C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
211#102—91no—
212#102—91no—
213H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—61no—
216C22×D3—181no—
217C3×D4—92no—
218C3×D4—181no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—362no—
232C2×C3×C4C3×D1211yescharacteristic
233C2×C3×C4C3×D1211yescharacteristic
234C2×C3×C4C22×C3×D311yescharacteristic
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—181no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241D12—32no—
242D12—61no—
243D12—92no—
244D12—181no—
245C4×D3—122no—
246C4×D3—362no—
247C3×C8—21no—
248C3×C8—21no—
249C3×C8—61no—
250C3×C8—61no—
251C3×C8—61no—
252C3×C8—61no—
253C3×C8—121no—
254C3×C8—121no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32#89711yescharacteristic
259C2×C32#90411yescharacteristic
260C2×C32#90811yescharacteristic
261C2×C32C3×C22⋊Q811yescharacteristic
262C2×C32C3×#13511yescharacteristic
263C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
264C3⋊D3—42no—
265C3×D3—122no—
266C2×D4—91no—
267C22×C4—181no—
268C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
269C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
270C4⋊C4—91no—
271C4⋊C4—91no—
272C4⋊C4—91no—
273C4⋊C4—181no—
274C4⋊C4—181no—
275C22⋊C4—181no—
276C42C2×(C3⋊D3)⋊C311yescharacteristic
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—61no—
280C22×C3—182no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3C2×C3×D1211yescharacteristic
284C2×D3—62no—
285C2×D3—121no—
286C2×D3—121no—
287C2×D3—182no—
288C2×D3—361no—
289C2×D3—361no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—31no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—61no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—61no—
300C3×C4—62no—
301C3×C4—61no—
302C3×C4—61no—
303C3×C4—121no—
304C3×C4—361no—
305C3×C4—361no—
306C3×C4—361no—
307C3×C4C3×#23711yescharacteristic
308C3×C4C3×D3×Q811yescharacteristic
309Dic3—121no—
310Dic3—121no—
311Dic3—121no—
312Dic3—361no—
313Dic3—361no—
314Dic3—361no—
315C32—21no—
316C32#484711yescharacteristic, complemented
317C32C3×#47811yescharacteristic, complemented
318C23—181no—
319D4—92no—
320D4—181no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—362no—
325C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
326C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
327C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
328C8—21no—
329C8—21no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—61no—
337C2×C3—61no—
338C2×C3—61no—
339C2×C3—362no—
340C2×C3C3×#89711yescharacteristic
341C2×C3C3×#90411yescharacteristic
342C2×C3C3×#90811yescharacteristic
343D3—122no—
344D3—362no—
345C22—182no—
346C22—361no—
347C22—361no—
348C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
349C4—21no—
350C4—21no—
351C4—361no—
352C4—361no—
353C4—361no—
354C4#3830⋊C311yescharacteristic
355C4Q8×(C3⋊D3)⋊C311yescharacteristic
356C3—31no—
357C3—31no—
358C3—61no—
359C3C3×#484711yescharacteristic
360C2—362no—
361C2#64106⋊C311yescharacteristic
362C2#64113⋊C311yescharacteristic
363C2#64117⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. H(3,𝔽3)×C4⋊C8C2semidirect
6. #63645⋊C3C2semidirect
7. #63644⋊C3C2semidirect
8. #10592546C3semidirect
22. #64079C2×C3semidirect
26. C32×C4⋊C8D3semidirect
27. C32×C4⋊C8C2×C3semidirect
30. #63645C2×C3semidirect
31. #63644C2×C3semidirect
210. C4⋊C8(C3⋊D3)⋊C3semidirect
213. H(3,𝔽3)#478semidirect
316. C32#4847semidirect
317. C32C3×#478semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊((C3⋊D3)⋊C3)2
#64079⋊C32
H(3,𝔽3)×C4⋊C82
#63648⋊C32
#63645⋊C32
#63644⋊C32
#105925463
C3×#48473