Finite Group Databasebeta

← #10592589⋊C3 #13142212

Subgroups — #10592589⋊C3 #13142212

Order 1728.

414 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 94 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#64097⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592589C311yesmaximal, characteristic, complemented
9C3×#4890—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×#634⋊C3—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64102C2×C311yescharacteristic
22#64097C2×C311yescharacteristic, complemented
23C3×#893—31no—
24C3×#888—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic, complemented
27C32×#107C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic
29#63641C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic
31#63634C2×C311yescharacteristic, complemented
32C3×#821—31no—
33C3×Dic3⋊C8—31no—
34C3×#809—31no—
35C3×#807—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)Q812yescomplemented
45C4×(C3⋊D3)⋊C3—42no—
46#634⋊C3—42no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#474—91no—
50#4890—31no—
51#4890—91no—
52C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
53C2×C4×C3⋊D3—21no—
54C2×#634—21no—
55C2×C3×D12—31no—
56C2×C3×C4×D3—61no—
57C2×C3×Dic6—61no—
58C2×C32×C8—21no—
59C2×C32×C8C2×D311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C32×C4⋊C4—21no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67#3752—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69#3750—21no—
70C4×C3⋊Dic3C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—31no—
74C3×Dic3⋊C4—61no—
75C3×C4×Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—22no—
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)C2×Q811yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3D812yes—
88C3×C4⋊Q8—91no—
89C3×C4×D4—91no—
90#893—31no—
91#893—91no—
92#888—31no—
93#888—91no—
94C3×#107—31no—
95C3×#107—31no—
96C3×#107—61no—
97C3×#107—91no—
98C3×#107C3×D311yescharacteristic
99C3×C4⋊C8—91no—
100C3×#102—91no—
101C3×#102—91no—
102#821—31no—
103#821—91no—
104Dic3⋊C8—31no—
105Dic3⋊C8—91no—
106#809—31no—
107#809—91no—
108#807—31no—
109#807—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—61no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3C3×D411yescharacteristic
123C2×C3⋊Dic3C3×D411yescharacteristic
124C4⋊(C3⋊D3)—21no—
125C4⋊(C3⋊D3)C3×Q812yescomplemented
126C4×C3⋊D3—42no—
127#634—42no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C3×D12—32no—
132C3×D12—61no—
133C3×C4×D3—122no—
134C3×Dic6—122no—
135C32×C8—22no—
136C32×C8—21no—
137C32×C8—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140#474—91noSylow
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)C2×D811yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144(C3⋊D3)⋊C3—42no—
145C2×C3×Q8—181no—
146C2×C3×D4—91no—
147C22×C3×C4—181no—
148C2×D12—31no—
149C2×D12—91no—
150C2×C4×D3—61no—
151C2×C4×D3—181no—
152C2×Dic6—61no—
153C2×Dic6—181no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—91no—
158C2×C3×C8C2×C3×D311yescharacteristic
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4C2×C3×D311yescharacteristic
168C3×C4⋊C4C2×C3×D311yescharacteristic
169C3×C22⋊C4—181no—
170C3×C42—91no—
171(C2×D3)⋊C4—61no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—91no—
175Dic3⋊C4—61no—
176Dic3⋊C4—181no—
177C4×Dic3—31no—
178C4×Dic3—91no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—91no—
181C22×C32—21no—
182C22×C32D3×D411yescharacteristic
183C22×C32C2×C3×D411yescharacteristic
184C2×C3⋊D3—22no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C2×C3×D3—62no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—81no—
197C32×C4—81no—
198C32×C4#23511yescharacteristic
199C32×C4D3×Q811yescharacteristic
200C32×C4C2×C3×Q811yescharacteristic
201C32×C4C3×#4111yescharacteristic
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3⋊Dic3C3×D812yes—
205C3×Dic3—32no—
206C3×Dic3—61no—
207C3×Dic3—121no—
208C4⋊Q8—91no—
209C4×D4—91no—
210#107—91no—
211#107(C3⋊D3)⋊C311yescharacteristic, complemented
212C4⋊C8—91no—
213#102—91no—
214#102—91no—
215H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
216C23×C3—181no—
217C22×D3—61no—
218C22×D3—181no—
219C3×Q8—362no—
220C3×D4—92no—
221C3×D4—181no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—121no—
234C2×C3×C4—121no—
235C2×C3×C4—181no—
236C2×C3×C4—362no—
237C2×C3×C4C22×C3×D311yescharacteristic
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—61no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C2×Dic3—181no—
244D12—32no—
245D12—61no—
246D12—92no—
247D12—181no—
248C4×D3—122no—
249C4×D3—362no—
250Dic6—122no—
251Dic6—362no—
252C3×C8—21no—
253C3×C8—61no—
254C3×C8—61no—
255C3×C8—62no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32#89611yescharacteristic
263C2×C32D3×D811yescharacteristic
264C2×C32#91811yescharacteristic
265C2×C32C3×C22⋊Q811yescharacteristic
266C2×C32C2×C3×D811yescharacteristic
267C2×C32C3×#13711yescharacteristic
268C3⋊D3—42no—
269C3×D3—122no—
270C2×Q8—181no—
271C2×D4—91no—
272C22×C4—181no—
273C2×C8—91no—
274C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
275C4⋊C4—91no—
276C4⋊C4—181no—
277C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
278C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
279C22⋊C4—181no—
280C42—91no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—61no—
284C22×C3—182no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3C3×D3×D411yescharacteristic
288C2×D3—62no—
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—182no—
292C2×D3—361no—
293C2×D3—361no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—92no—
303C3×C4—121no—
304C3×C4—121no—
305C3×C4—121no—
306C3×C4—121no—
307C3×C4—181no—
308C3×C4—241no—
309C3×C4—241no—
310C3×C4—361no—
311C3×C4C3×#23511yescharacteristic
312C3×C4C3×D3×Q811yescharacteristic
313Dic3—32no—
314Dic3—61no—
315Dic3—92no—
316Dic3—121no—
317Dic3—181no—
318Dic3—361no—
319C32—21no—
320C32#489011yescharacteristic, complemented
321C32C3×#47411yescharacteristic, complemented
322C23—181no—
323Q8—362no—
324D4—92no—
325D4—181no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—362no—
332C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
333C8—21no—
334C8—181no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—362no—
345C2×C3C3×#89611yescharacteristic
346C2×C3C3×D3×D811yescharacteristic
347C2×C3C3×#91811yescharacteristic
348D3—122no—
349D3—362no—
350C22—182no—
351C22—361no—
352C22—361no—
353C22D4×(C3⋊D3)⋊C311yescharacteristic
354C4—41no—
355C4—41no—
356C4—92no—
357C4—181no—
358C4—361no—
359C4#3828⋊C311yescharacteristic
360C4Q8×(C3⋊D3)⋊C311yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3C3×#489011yescharacteristic
365C2—362no—
366C2#64105⋊C311yescharacteristic
367C2D8×(C3⋊D3)⋊C311yescharacteristic
368C2#64127⋊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×#474
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. #64097⋊C3C2semidirect
3. H(3,𝔽3)×#107C2semidirect
5. #63641⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592589C3semidirect
22. #64097C2×C3semidirect
26. C32×#107D3semidirect
27. C32×#107C2×C3semidirect
29. #63641C2×C3semidirect
31. #63634C2×C3semidirect
44. C4⋊((C3⋊D3)⋊C3)Q8semidirect
125. C4⋊(C3⋊D3)C3×Q8semidirect
211. #107(C3⋊D3)⋊C3semidirect
215. H(3,𝔽3)#474semidirect
320. C32#4890semidirect
321. C32C3×#474semidirect

Maximal subgroups

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

subgroupindex
#64102⋊C32
#64097⋊C32
H(3,𝔽3)×#1072
#63648⋊C32
#63641⋊C32
#63636⋊C32
#63634⋊C32
#105925893
C3×#48903