Finite Group Databasebeta

← #10592590⋊C3 #13142214

Subgroups — #10592590⋊C3 #13142214

Order 1728.

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

normal subgroups only

This group has 393 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#64102⋊C3C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic
7#63635⋊C3C211yesmaximal, characteristic, complemented
8#10592590C311yesmaximal, characteristic, complemented
9C3×#4891—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64102C2×C311yescharacteristic
22#64099C2×C311yescharacteristic, complemented
23C3×#893—31no—
24C3×#890—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic, complemented
27C32×#107C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic
29#63641C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic
31#63635C2×C311yescharacteristic, complemented
32C3×#821—31no—
33C3×Dic3⋊C8—31no—
34C3×#809—31no—
35C3×#808—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yes—
45C4×(C3⋊D3)⋊C3—42no—
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48C3×#478—91no—
49#4891—31no—
50#4891—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×C8C2×D311yescharacteristic
57C2×C32×C8C22×C311yescharacteristic
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C22×C311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64#3752—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68C4×C3⋊Dic3C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×(C2×D3)⋊C4—61no—
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)C2×D411yescharacteristic
77C2×(C3⋊D3)⋊C3—22no—
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C2×Q811yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87C3×#125—91no—
88C3×C4×D4—91no—
89#893—31no—
90#893—91no—
91#890—31no—
92#890—91no—
93C3×#107—31no—
94C3×#107—31no—
95C3×#107—61no—
96C3×#107C3×D311yescharacteristic
97C3×#106—91no—
98C3×C4⋊C8—91no—
99C3×#102—91no—
100C3×#102—91no—
101#821—31no—
102#821—91no—
103Dic3⋊C8—31no—
104Dic3⋊C8—91no—
105#809—31no—
106#809—91no—
107#808—31no—
108#808—91no—
109C22×C3⋊D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3C3×D411yescharacteristic
122C2×C3⋊Dic3C3×D411yescharacteristic
123C4⋊(C3⋊D3)—21no—
124C4⋊(C3⋊D3)C3×Q812yes—
125C4×C3⋊D3—42no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C3×D12—32no—
130C3×D12—61no—
131C3×C4×D3—122no—
132C32×C8—22no—
133C32×C8—21no—
134C32×C8—21no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—61no—
137#478—91noSylow
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)#13511yescharacteristic
140C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
141(C3⋊D3)⋊C3—42no—
142C2×C3×D4—91no—
143C22×C3×C4—181no—
144C2×D12—31no—
145C2×D12—91no—
146C2×C4×D3—61no—
147C2×C4×D3—181no—
148C2×C3×C8—31no—
149C2×C3×C8—31no—
150C2×C3×C8—61no—
151C2×C3×C8—91no—
152C2×C3×C8C2×C3×D311yescharacteristic
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4C2×C3×D311yescharacteristic
163C3×C4⋊C4C2×C3×D311yescharacteristic
164C3×C22⋊C4—181no—
165C3×C42—91no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—181no—
168C4⋊Dic3—31no—
169C4⋊Dic3—61no—
170C4⋊Dic3—91no—
171C4⋊Dic3—181no—
172Dic3⋊C4—61no—
173Dic3⋊C4—181no—
174C4×Dic3—31no—
175C4×Dic3—91no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—91no—
178C22×C32—21no—
179C22×C32D3×D411yescharacteristic
180C22×C32C2×C3×D411yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3×D3—62no—
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4—81no—
194C32×C4—81no—
195C32×C4#23511yescharacteristic
196C32×C4D3×Q811yescharacteristic
197C32×C4C2×C3×Q811yescharacteristic
198C32×C4C3×#4111yescharacteristic
199C3⋊Dic3—21no—
200C3⋊Dic3—21no—
201C3⋊Dic3—41no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—121no—
205#125—91no—
206C4×D4—91no—
207#107(C3⋊D3)⋊C311yescharacteristic, complemented
208#106—91no—
209C4⋊C8—91no—
210#102—91no—
211#102—91no—
212H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
213C23×C3—181no—
214C22×D3—61no—
215C22×D3—181no—
216C3×D4—92no—
217C3×D4—181no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—121no—
230C2×C3×C4—121no—
231C2×C3×C4—181no—
232C2×C3×C4—362no—
233C2×C3×C4C22×C3×D311yescharacteristic
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—61no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—181no—
240D12—32no—
241D12—61no—
242D12—92no—
243D12—181no—
244C4×D3—122no—
245C4×D3—362no—
246C3×C8—21no—
247C3×C8—61no—
248C3×C8—61no—
249C3×C8—62no—
250C3×C8—181no—
251C3⋊C8—61no—
252C3⋊C8—181no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32#89611yescharacteristic
257C2×C32#91111yescharacteristic
258C2×C32#91911yescharacteristic
259C2×C32C3×C22⋊Q811yescharacteristic
260C2×C32C3×#13511yescharacteristic
261C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
262C3⋊D3—42no—
263C3×D3—122no—
264C2×D4—91no—
265C22×C4—181no—
266C2×C8—91no—
267C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
268C4⋊C4—91no—
269C4⋊C4—181no—
270C4⋊C4—181no—
271C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
272C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
273C22⋊C4—181no—
274C42—91no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—61no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3C3×D3×D411yescharacteristic
282C2×D3—62no—
283C2×D3—121no—
284C2×D3—121no—
285C2×D3—182no—
286C2×D3—361no—
287C2×D3—361no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—121no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—121no—
300C3×C4—181no—
301C3×C4—181no—
302C3×C4—241no—
303C3×C4—241no—
304C3×C4—361no—
305C3×C4C3×#23511yescharacteristic
306C3×C4C3×D3×Q811yescharacteristic
307Dic3—61no—
308Dic3—61no—
309Dic3—121no—
310Dic3—181no—
311Dic3—181no—
312Dic3—361no—
313C32—21no—
314C32#489111yescharacteristic, complemented
315C32C3×#47811yescharacteristic, complemented
316C23—181no—
317D4—92no—
318D4—181no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—181no—
324C2×C4—362no—
325C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
326C8—21no—
327C8—181no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—362no—
338C2×C3C3×#89611yescharacteristic
339C2×C3C3×#91111yescharacteristic
340C2×C3C3×#91911yescharacteristic
341D3—122no—
342D3—362no—
343C22—182no—
344C22—361no—
345C22—361no—
346C22D4×(C3⋊D3)⋊C311yescharacteristic
347C4—41no—
348C4—41no—
349C4—181no—
350C4—181no—
351C4—361no—
352C4#3828⋊C311yescharacteristic
353C4Q8×(C3⋊D3)⋊C311yescharacteristic
354C3—31no—
355C3—31no—
356C3—61no—
357C3C3×#489111yescharacteristic
358C2—362no—
359C2#64105⋊C311yescharacteristic
360C2#64120⋊C311yescharacteristic
361C2#64128⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #64099⋊C3C2semidirect
3. H(3,𝔽3)×#107C2semidirect
5. #63641⋊C3C2semidirect
7. #63635⋊C3C2semidirect
8. #10592590C3semidirect
22. #64099C2×C3semidirect
26. C32×#107D3semidirect
27. C32×#107C2×C3semidirect
29. #63641C2×C3semidirect
31. #63635C2×C3semidirect
207. #107(C3⋊D3)⋊C3semidirect
212. H(3,𝔽3)#478semidirect
314. C32#4891semidirect
315. C32C3×#478semidirect

Maximal subgroups

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

subgroupindex
#64102⋊C32
#64099⋊C32
H(3,𝔽3)×#1072
#63648⋊C32
#63641⋊C32
#63636⋊C32
#63635⋊C32
#105925903
C3×#48913