Finite Group Databasebeta

← #10592510⋊C3 #13141922

Subgroups — #10592510⋊C3 #13141922

Order 1728.

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

normal subgroups only

This group has 406 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#64155⋊C3C211yesmaximal, characteristic
2C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic
6#63645⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592510C311yesmaximal, characteristic, complemented
9C3×#4811—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19#3750⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64155C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic
23C3×#946—31no—
24C3×D3×C4⋊C4—31no—
25C32×#103—21no—
26C32×#103D311yescharacteristic, complemented
27C32×#103C2×C311yescharacteristic, complemented
28#63662C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic
30#63645C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×#835—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#818—31no—
35C3×#807—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—42no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50C3×#484—91no—
51#4811—31no—
52#4811—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C22×C311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—61no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C22×C311yescharacteristic
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66#3752—21no—
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69#3750—21no—
70#3750—21no—
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×Dic3⋊C4—61no—
76C3×Dic3⋊C4—61no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3Q1612yes—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)#4111yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×C22⋊Q8—91no—
90C2×C3×C4⋊C4—91no—
91#946—31no—
92#946—91no—
93D3×C4⋊C4—31no—
94D3×C4⋊C4—91no—
95C3×#107—91no—
96C3×#107—91no—
97C3×#103—31no—
98C3×#103—31no—
99C3×#103—61no—
100C3×#103—91no—
101C3×#103C3×D311yescharacteristic
102C3×C22⋊C8—91no—
103#835—31no—
104#835—91no—
105(C2×D3)⋊C8—31no—
106(C2×D3)⋊C8—91no—
107#818—31no—
108#818—91no—
109#807—31no—
110#807—91no—
111C22×C3⋊D3C3×D411yescharacteristic
112C22×C3×D3—31no—
113C32×Q8—22no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—41no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C23×C311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3C3×D411yescharacteristic
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C4×C3⋊D3—42no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—122no—
140C32×C8—21no—
141C32×C8—21no—
142C32×C8—41no—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—61no—
145#484—91noSylow
146C2×H(3,𝔽3)#12311yescharacteristic
147C2×H(3,𝔽3)C2×Q1611yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149(C3⋊D3)⋊C3—22no—
150C2×C3×Q8—31no—
151C2×C3×Q8—31no—
152C2×C3×Q8—61no—
153C2×C3×Q8C2×C3×D311yescharacteristic
154C22×C3×C4—91no—
155C22×C3×C4—181no—
156C2×C4×D3—31no—
157C2×C4×D3—61no—
158C2×C4×D3—91no—
159C2×C4×D3—181no—
160C2×C3×C8—31no—
161C2×C3×C8—31no—
162C2×C3×C8—61no—
163C2×C3×C8—91no—
164C2×C3×C8C2×C3×D311yescharacteristic
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—181no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4C2×C3×D311yescharacteristic
173C3×C22⋊C4—181no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—91no—
179C4⋊Dic3—91no—
180Dic3⋊C4—61no—
181Dic3⋊C4—61no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32D3×D411yescharacteristic
188C22×C32C2×C3×D411yescharacteristic
189C2×C3⋊D3—21no—
190C2×C3⋊D3—21no—
191C2×C3⋊D3C3×Q1612yes—
192C2×C3×D3—32no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—42no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4—41no—
201C32×C4—41no—
202C32×C4—81no—
203C32×C4#23511yescharacteristic
204C32×C4#23711yescharacteristic
205C32×C4C3×#4111yescharacteristic
206C32×C4C3×#4111yescharacteristic
207C3⋊Dic3—21no—
208C3⋊Dic3—41no—
209C3⋊Dic3—41no—
210C3×Dic3—61no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C22⋊Q8—91no—
214C2×C4⋊C4—91no—
215#107—91no—
216#107—91no—
217#103—91no—
218#103(C3⋊D3)⋊C311yescharacteristic, complemented
219C22⋊C8—91no—
220H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
221C23×C3—91no—
222C22×D3—31no—
223C22×D3—91no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—61no—
227C3×Q8—61no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×Q8—62no—
231C3×Q8—121no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—91no—
242C2×C3×C4—121no—
243C2×C3×C4—121no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—362no—
249C2×C3×C4C22×C3×D311yescharacteristic
250C2×Dic3—31no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—91no—
254C2×Dic3—181no—
255C2×Dic3—181no—
256C4×D3—61no—
257C4×D3—61no—
258C4×D3—122no—
259C4×D3—181no—
260C4×D3—181no—
261C4×D3—362no—
262C3×C8—21no—
263C3×C8—61no—
264C3×C8—61no—
265C3×C8—121no—
266C3×C8—181no—
267C3⋊C8—61no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32#88311yescharacteristic
273C2×C32#91411yescharacteristic
274C2×C32D3×Q1611yescharacteristic
275C2×C32C3×#12311yescharacteristic
276C2×C32C2×C3×Q1611yescharacteristic
277C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
278C3⋊D3—22no—
279C3×D3—62no—
280C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
281C22×C4—91no—
282C22×C4—181no—
283C2×C8—91no—
284C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—181no—
288C4⋊C4—181no—
289C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
290C22⋊C4—181no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—61no—
294C22×C3—92no—
295C22×C3—181no—
296C22×C3—181no—
297C22×C3C3×D3×D411yescharacteristic
298C2×D3—32no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—92no—
302C2×D3—181no—
303C2×D3—181no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—61no—
311C3×C4—61no—
312C3×C4—121no—
313C3×C4—121no—
314C3×C4—121no—
315C3×C4—121no—
316C3×C4—122no—
317C3×C4—181no—
318C3×C4—241no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4C3×#23511yescharacteristic
322C3×C4C3×#23711yescharacteristic
323Dic3—61no—
324Dic3—121no—
325Dic3—121no—
326Dic3—181no—
327Dic3—361no—
328Dic3—361no—
329C32—21no—
330C32#481111yescharacteristic, complemented
331C32C3×#48411yescharacteristic, complemented
332C23—91no—
333Q8—21no—
334Q8—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—362no—
343C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
344C8—21no—
345C8—181no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—61no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—182no—
356C2×C3C3×#88311yescharacteristic
357C2×C3C3×#91411yescharacteristic
358C2×C3C3×D3×Q1611yescharacteristic
359D3—62no—
360D3—182no—
361C22—92no—
362C22—181no—
363C22—181no—
364C22D4×(C3⋊D3)⋊C311yescharacteristic
365C4—41no—
366C4—41no—
367C4—181no—
368C4—361no—
369C4—361no—
370C4#3828⋊C311yescharacteristic
371C4#3830⋊C311yescharacteristic
372C3—31no—
373C3—31no—
374C3—61no—
375C3C3×#481111yescharacteristic
376C2—182no—
377C2#64092⋊C311yescharacteristic
378C2#64123⋊C311yescharacteristic
379C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#484
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#103C2semidirect
4. #63662⋊C3C2semidirect
6. #63645⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592510C3semidirect
26. C32×#103D3semidirect
27. C32×#103C2×C3semidirect
28. #63662C2×C3semidirect
30. #63645C2×C3semidirect
31. #63634C2×C3semidirect
218. #103(C3⋊D3)⋊C3semidirect
220. H(3,𝔽3)#484semidirect
330. C32#4811semidirect
331. C32C3×#484semidirect

Maximal subgroups

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

subgroupindex
#64155⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63647⋊C32
#63645⋊C32
#63634⋊C32
#105925103
C3×#48113