Finite Group Databasebeta

← #10592494⋊C3 #13141906

Subgroups — #10592494⋊C3 #13141906

Order 1728.

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

normal subgroups only

This group has 420 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#64153⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic
3C2×#3746⋊C3C211yesmaximal, characteristic
4#64096⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic
6#63641⋊C3C211yesmaximal, characteristic
7#63637⋊C3C211yesmaximal, characteristic
8#10592494C311yesmaximal, characteristic, complemented
9C3×#4795—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3756⋊C3—22no—
16#3750⋊C3—21no—
17#3750⋊C3—21no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3746⋊C3—22no—
23#64153C2×C311yescharacteristic
24C2×#3756C2×C311yescharacteristic
25C2×#3746C2×C311yescharacteristic
26#64096C2×C311yescharacteristic
27C3×#944—31no—
28C2×C3×#216—31no—
29C2×C3×Dic12—31no—
30C3×#887—31no—
31C32×#103—21no—
32C32×#103D311yescharacteristic
33C32×#103C2×C311yescharacteristic
34#63641C2×C311yescharacteristic
35#63637C2×C311yescharacteristic
36C3×Dic3⋊C8—31no—
37C3×#810—31no—
38Q8×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47#634⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51C8×H(3,𝔽3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53C3×#462—91no—
54#4795—31no—
55#4795—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C22×C311yescharacteristic
59C2×#634C22×C311yescharacteristic
60C2×#634C22×C311yescharacteristic
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C22×C311yescharacteristic
69#3756—22no—
70#3750—21no—
71#3750—21no—
72#3750—21no—
73C4×C3⋊Dic3—21no—
74C4×C3⋊Dic3C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76#3746—22no—
77C3×#216—62no—
78C3×Dic3⋊C4—61no—
79C3×Dic3⋊C4—61no—
80C3×Dic3⋊C4—61no—
81C3×C4×Dic3—31no—
82C3×C4×Dic3—61no—
83C2×C3×C3⋊C8—31no—
84C3×Dic12—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)—41no—
88C4×H(3,𝔽3)C2×D411yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—41no—
93(C3⋊Dic3)⋊C3Q1612yescomplemented
94C2×C3×Q16—91no—
95C2×C3×Q16—91no—
96C3×C4⋊Q8—91no—
97C3×C4×Q8—91no—
98#944—31no—
99#944—91no—
100C2×#216—31no—
101C2×#216—91no—
102C2×Dic12—31no—
103C2×Dic12—91no—
104#887—31no—
105#887—91no—
106C3×C4⋊C8—91no—
107C3×#103—31no—
108C3×#103—31no—
109C3×#103—61no—
110C3×#103—91no—
111C3×#103C3×D311yescharacteristic
112Dic3⋊C8—31no—
113Dic3⋊C8—91no—
114#810—31no—
115#810—91no—
116C32×Q8—22no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—41no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C23×C311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3C3×D411yescharacteristic
134C2×C3⋊Dic3C3×D411yescharacteristic
135#634—21no—
136#634—21no—
137#634—21no—
138#634C3×D412yes—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3×Dic6—32no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C32×C8—21no—
148C32×C8—21no—
149C32×C8—41no—
150C3⋊(C3⋊C8)—21no—
151C3×C3⋊C8—61no—
152#462—91noSylow
153C2×H(3,𝔽3)#12111yescharacteristic
154C2×H(3,𝔽3)C2×Q1611yescharacteristic
155C2×H(3,𝔽3)#13711yescharacteristic
156C2×C3×Q8—31no—
157C2×C3×Q8—31no—
158C2×C3×Q8—61no—
159C2×C3×Q8—91no—
160C2×C3×Q8—91no—
161C2×C3×Q8C2×C3×D311yescharacteristic
162C2×Dic6—31no—
163C2×Dic6—31no—
164C2×Dic6—91no—
165C2×Dic6—91no—
166C3×Q16—182no—
167C3×Q16—182no—
168C2×C3×C8—31no—
169C2×C3×C8—31no—
170C2×C3×C8—61no—
171C2×C3×C8—91no—
172C2×C3×C8C2×C3×D311yescharacteristic
173C3×C4⋊C4—31no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—181no—
177C3×C4⋊C4—181no—
178C3×C4⋊C4—181no—
179C3×C4⋊C4C2×C3×D311yescharacteristic
180C3×C42—91no—
181C3×C42—181no—
182#216—62no—
183#216—182no—
184Dic3⋊C4—61no—
185Dic3⋊C4—61no—
186Dic3⋊C4—61no—
187Dic3⋊C4—181no—
188Dic3⋊C4—181no—
189Dic3⋊C4—181no—
190C4×Dic3—31no—
191C4×Dic3—61no—
192C4×Dic3—91no—
193C4×Dic3—181no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—91no—
196Dic12—62no—
197Dic12—182no—
198C22×C32—21no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×C3×D411yescharacteristic
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—42no—
204C32×C4—41no—
205C32×C4—41no—
206C32×C4—41no—
207C32×C4—41no—
208C32×C4—81no—
209C32×C4#23511yescharacteristic
210C32×C4D3×D411yescharacteristic
211C32×C4C2×C3×D411yescharacteristic
212C32×C4C3×#4111yescharacteristic
213C3⋊Dic3—22no—
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3⋊Dic3C3×Q1612yescomplemented
217C3×Dic3—32no—
218C3×Dic3—62no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C2×Q16—91no—
222C2×Q16—91no—
223C4⋊Q8—91no—
224C4×Q8—91no—
225C4⋊C8—91no—
226#103—91no—
227#103(C3⋊D3)⋊C311yescharacteristic
228H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—61no—
232C3×Q8—61no—
233C3×Q8—61no—
234C3×Q8—61no—
235C3×Q8—62no—
236C3×Q8—92no—
237C3×Q8—121no—
238C3×Q8—181no—
239C3×Q8—181no—
240C3×Q8—181no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—121no—
253C2×C3×C4—121no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4C22×C3×D311yescharacteristic
257C2×Dic3—31no—
258C2×Dic3—31no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—91no—
262C2×Dic3—91no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265Dic6—32no—
266Dic6—61no—
267Dic6—61no—
268Dic6—61no—
269Dic6—92no—
270Dic6—181no—
271Dic6—181no—
272Dic6—181no—
273C3×C8—21no—
274C3×C8—61no—
275C3×C8—61no—
276C3×C8—121no—
277C3×C8—181no—
278C3⋊C8—61no—
279C3⋊C8—181no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32#88411yescharacteristic
284C2×C32#91511yescharacteristic
285C2×C32D3×Q1611yescharacteristic
286C2×C32C3×#12111yescharacteristic
287C2×C32C2×C3×Q1611yescharacteristic
288C2×C32C3×#13711yescharacteristic
289C2×Q8—91no—
290C2×Q8—91no—
291C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
292Q16—182no—
293Q16—182no—
294C2×C8—91no—
295C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
296C4⋊C4—181no—
297C4⋊C4—181no—
298C4⋊C4—181no—
299C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
300C42—91no—
301C42—181no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—61no—
305C22×C3C3×D3×D411yescharacteristic
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—92no—
315C3×C4—121no—
316C3×C4—121no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—122no—
320C3×C4—181no—
321C3×C4—182no—
322C3×C4—241no—
323C3×C4—361no—
324C3×C4C3×#23511yescharacteristic
325C3×C4C3×D3×D411yescharacteristic
326Dic3—32no—
327Dic3—62no—
328Dic3—61no—
329Dic3—92no—
330Dic3—121no—
331Dic3—182no—
332Dic3—181no—
333Dic3—361no—
334C32—21no—
335C32#479511yescharacteristic, complemented
336C32C3×#46211yescharacteristic, complemented
337Q8—21no—
338Q8—21no—
339Q8—92no—
340Q8—181no—
341Q8—181no—
342Q8—181no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—91no—
346C2×C4—91no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
350C8—21no—
351C8—181no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3C3×#88411yescharacteristic
362C2×C3C3×#91511yescharacteristic
363C2×C3C3×D3×Q1611yescharacteristic
364C22D4×(C3⋊D3)⋊C311yescharacteristic
365C4—41no—
366C4—41no—
367C4—92no—
368C4—181no—
369C4—182no—
370C4—361no—
371C4#3828⋊C311yescharacteristic
372C4D4×(C3⋊D3)⋊C311yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3C3×#479511yescharacteristic
377C2#64093⋊C311yescharacteristic
378C2#64124⋊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×H(3,𝔽3)C2×D4
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#462
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
8. #10592494C3semidirect
93. (C3⋊Dic3)⋊C3Q16semidirect
216. C3⋊Dic3C3×Q16semidirect
228. H(3,𝔽3)#462semidirect
335. C32#4795semidirect
336. C32C3×#462semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
C2×#3756⋊C32
C2×#3746⋊C32
#64096⋊C32
H(3,𝔽3)×#1032
#63641⋊C32
#63637⋊C32
#105924943
C3×#47953