Finite Group Databasebeta

← #10592465⋊C3 #13141781

Subgroups — #10592465⋊C3 #13141781

Order 1728.

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

normal subgroups only

This group has 407 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
1D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#63645⋊C3C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#63635⋊C3C211yesmaximal, characteristic, complemented
8#10592465C311yesmaximal, characteristic, complemented
9C3×#4766—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#3757⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21D4×C3⋊Dic3C2×C311yescharacteristic
22#64099C2×C311yescharacteristic, complemented
23C3×D4×Dic3—31no—
24C3×#890—31no—
25C32×#102—21no—
26C32×#102D311yescharacteristic
27C32×#102C2×C311yescharacteristic
28#63659C2×C311yescharacteristic
29#63645C2×C311yescharacteristic, complemented
30#63641C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C3×#832—31no—
33C3×#818—31no—
34C3×Dic3⋊C8—31no—
35C3×#808—31no—
36C23×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)Q812yes—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—42no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48C3×#478—91no—
49#4766—31no—
50#4766—91no—
51C2×C32×D4—21no—
52C2×C32×D4C2×D311yescharacteristic
53C2×C32×D4C22×C311yescharacteristic
54C22×C3⋊Dic3—21no—
55C22×C3×Dic3—61no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C22×C311yescharacteristic
62#3757—21no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)C22×C311yescharacteristic
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66#3750—21no—
67C4×C3⋊Dic3C22×C311yescharacteristic
68C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
69C3×#217—61no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—31no—
75C2×C3×C3⋊C8—31no—
76C22×H(3,𝔽3)—22no—
77C22×H(3,𝔽3)—41no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)C2×Q811yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87C3×#125—91no—
88C3×C4×D4—91no—
89D4×Dic3—31no—
90D4×Dic3—91no—
91#890—31no—
92#890—91no—
93C3×#107—91no—
94C3×#106—91no—
95C3×C4⋊C8—91no—
96C3×#102—31no—
97C3×#102—31no—
98C3×#102—61no—
99C3×#102—91no—
100C3×#102C3×D311yescharacteristic
101#832—31no—
102#832—91no—
103#818—31no—
104#818—91no—
105Dic3⋊C8—31no—
106Dic3⋊C8—91no—
107#808—31no—
108#808—91no—
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—41no—
112C32×D4—21no—
113C32×D4—21no—
114C32×D4—22no—
115C32×D4—41no—
116C32×D4Dic612yes—
117C32×D4C3×Q812yes—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C23×C311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—42no—
127C2×C3⋊Dic3C3×D411yescharacteristic
128C2×C3⋊Dic3C3×D411yescharacteristic
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—122no—
134C32×C8—21no—
135C32×C8—21no—
136C32×C8—41no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139#478—91noSylow
140C2×H(3,𝔽3)—42no—
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)#13511yescharacteristic
143C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
144C2×C3×D4—31no—
145C2×C3×D4—31no—
146C2×C3×D4—61no—
147C2×C3×D4C2×C3×D311yescharacteristic
148C22×C3×C4—181no—
149C22×Dic3—61no—
150C22×Dic3—181no—
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—61no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4—181no—
163C3×C4⋊C4C2×C3×D311yescharacteristic
164C3×C22⋊C4—181no—
165C3×C42—91no—
166#217—61no—
167#217—181no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174Dic3⋊C4—61no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—91no—
180C22×C32—22no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—42no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—81no—
189C22×C32—81no—
190C22×C32D3×D411yescharacteristic
191C22×C32C2×C3×D411yescharacteristic
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—81no—
197C32×C4C2×Dic611yescharacteristic
198C32×C4#23711yescharacteristic
199C32×C4C2×C3×Q811yescharacteristic
200C32×C4C3×#4111yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209#125—91no—
210C4×D4—91no—
211#107—91no—
212#106—91no—
213C4⋊C8—91no—
214#102—91no—
215#102(C3⋊D3)⋊C311yescharacteristic
216H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
217C23×C3—21no—
218C23×C3—61no—
219C23×C3—61no—
220C23×C3—121no—
221C3×D4—21no—
222C3×D4—32no—
223C3×D4—32no—
224C3×D4—61no—
225C3×D4—61no—
226C3×D4—62no—
227C3×D4—121no—
228C3×D4C3×Dic612yes—
229C2×C3×C4—21no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—121no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—362no—
241C2×C3×C4C22×C3×D311yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—91no—
247C2×Dic3—91no—
248C2×Dic3—122no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—362no—
252C3×C8—21no—
253C3×C8—61no—
254C3×C8—61no—
255C3×C8—121no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—42no—
263C2×C32—42no—
264C2×C32—82no—
265C2×C32#87811yescharacteristic
266C2×C32#91211yescharacteristic
267C2×C32#91411yescharacteristic
268C2×C32C3×C22⋊Q811yescharacteristic
269C2×C32C3×#13511yescharacteristic
270C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
271C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
272C22×C4—181no—
273C2×C8—91no—
274C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4—181no—
279C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
280C22⋊C4—181no—
281C42—91no—
282C22×C3—22no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—61no—
290C22×C3—122no—
291C22×C3—121no—
292C22×C3—121no—
293C22×C3—121no—
294C22×C3—121no—
295C22×C3—241no—
296C22×C3—241no—
297C22×C3C3×D3×D411yescharacteristic
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—41no—
303C3×C4—61no—
304C3×C4—61no—
305C3×C4—121no—
306C3×C4—121no—
307C3×C4—181no—
308C3×C4—181no—
309C3×C4—241no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4C2×C3×Dic611yescharacteristic
313C3×C4C3×#23711yescharacteristic
314Dic3—61no—
315Dic3—61no—
316Dic3—121no—
317Dic3—121no—
318Dic3—181no—
319Dic3—181no—
320Dic3—361no—
321Dic3—361no—
322C32—21no—
323C32#476611yescharacteristic, complemented
324C32C3×#47811yescharacteristic, complemented
325C23—21no—
326D4—21no—
327D4#634⋊C312yes—
328C2×C4—21no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—362no—
334C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
335C8—21no—
336C8—181no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—42no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—122no—
348C2×C3—122no—
349C2×C3—242no—
350C2×C3C3×#87811yescharacteristic
351C2×C3C3×#91211yescharacteristic
352C2×C3C3×#91411yescharacteristic
353C22—22no—
354C22—41no—
355C22—41no—
356C22D4×(C3⋊D3)⋊C311yescharacteristic
357C4—41no—
358C4—181no—
359C4—181no—
360C4—361no—
361C4—361no—
362C4C2×#634⋊C311yescharacteristic
363C4#3830⋊C311yescharacteristic
364C3—31no—
365C3—31no—
366C3—61no—
367C3C3×#476611yescharacteristic
368C2—42no—
369C2#64087⋊C311yescharacteristic
370C2#64121⋊C311yescharacteristic
371C2#64123⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×H(3,𝔽3)C22
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #64099⋊C3C2semidirect
5. #63645⋊C3C2semidirect
6. #63641⋊C3C2semidirect
7. #63635⋊C3C2semidirect
8. #10592465C3semidirect
22. #64099C2×C3semidirect
29. #63645C2×C3semidirect
30. #63641C2×C3semidirect
31. #63635C2×C3semidirect
216. H(3,𝔽3)#478semidirect
323. C32#4766semidirect
324. C32C3×#478semidirect

Maximal subgroups

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

subgroupindex
D4×(C3⋊Dic3)⋊C32
#64099⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63645⋊C32
#63641⋊C32
#63635⋊C32
#105924653
C3×#47663