Finite Group Databasebeta

← #10592454⋊C3 #13141700

Subgroups — #10592454⋊C3 #13141700

Order 1728.

388 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 80 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#64151⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3#64118⋊C3C211yesmaximal, characteristic
4#64116⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
6#63651⋊C3C211yesmaximal, characteristic, complemented
7#63649⋊C3C211yesmaximal, characteristic, complemented
8#10592454C311yesmaximal, characteristic, complemented
9C3×#4755—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14M16×H(3,𝔽3)C2211yescharacteristic
15#3755⋊C3—21no—
16#3756⋊C3—21no—
17#3756⋊C3—21no—
18#3756⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3744⋊C3—21no—
22#3743⋊C3—21no—
23C8×(C3⋊D3)⋊C3—21no—
24#3746⋊C3—21no—
25C2×#3756C2×C311yescharacteristic, complemented
26#64151C2×C311yescharacteristic
27#64118C2×C311yescharacteristic
28#64116C2×C311yescharacteristic
29C2×C3×#216—31no—
30C3×#942—31no—
31C3×#909—31no—
32C3×#907—31no—
33C32×#101—21no—
34C32×#101D311yescharacteristic, complemented
35C32×#101C2×C311yescharacteristic, complemented
36#63651C2×C311yescharacteristic, complemented
37#63649C2×C311yescharacteristic, complemented
38C3×#824—31no—
39C3×#822—31no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
47C4×(C3⋊D3)⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D411yescharacteristic
51C8×H(3,𝔽3)—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56C3×#473—91no—
57#4755—31no—
58#4755—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C311yescharacteristic
62#3828C22×C311yescharacteristic
63C2×#634C22×C311yescharacteristic
64C3×#235—31no—
65C2×C3×Dic6—31no—
66C32×M16—21no—
67C32×M16—21no—
68C32×M16C2×D311yescharacteristic
69C32×M16C2×D311yescharacteristic
70C32×M16C22×C311yescharacteristic
71C32×M16C22×C311yescharacteristic
72#3756—21no—
73#3756—21no—
74#3756—21no—
75#3755—21no—
76#3748C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3746—21no—
79#3744—21no—
80#3743—21no—
81C8×C3⋊D3—21no—
82C3×#216—61no—
83C3×#216—61no—
84C3×#216—61no—
85C3×#215—61no—
86C3×#208—31no—
87C2×C3×C3⋊C8—31no—
88C3×Dic12—61no—
89C3×#204—61no—
90C3×ΓL(1,𝔽25)—61no—
91C3×D3×C8—61no—
92C22×H(3,𝔽3)#4111yescharacteristic
93C2×(C3⋊D3)⋊C3—21no—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C3×#137—91no—
100C3×#137—91no—
101C2×C3×Q16—91no—
102C3×#131—91no—
103C2×#216—31no—
104C2×#216—91no—
105#942—31no—
106#942—91no—
107#909—31no—
108#909—91no—
109#907—31no—
110#907—91no—
111C3×#108—91no—
112C3×#101—31no—
113C3×#101—31no—
114C3×#101—61no—
115C3×#101—91no—
116C3×#101C3×D311yescharacteristic
117#824—31no—
118#824—91no—
119#822—31no—
120#822—91no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—41no—
126C32×Q8—41no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C23×C311yescharacteristic
133#638—21no—
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)C3×D411yescharacteristic
136C4×C3⋊D3—21no—
137#634—21no—
138#634—21no—
139#634C3×D411yescharacteristic
140C3×#67—61no—
141C2×C3×Dic3—61no—
142C3×D12—31no—
143C3×C4×D3—61no—
144C3×Dic6—31no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C32×C8—21no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—41no—
152C32×C8—41no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)C3×D411yescharacteristic
155C3⋊(C3⋊C8)C3×D411yescharacteristic
156C3×C3⋊C8—31no—
157C3×C3⋊C8—31no—
158C3×C3⋊C8—61no—
159#473—91noSylow
160C2×H(3,𝔽3)—21no—
161C2×H(3,𝔽3)#12111yescharacteristic
162(C3⋊D3)⋊C3—41no—
163C3×#41—91no—
164C2×C3×Q8—31no—
165C2×C3×Q8—31no—
166C2×C3×Q8—61no—
167C2×C3×Q8—91no—
168C2×C3×Q8C2×C3×D311yescharacteristic
169#235—31no—
170#235—91no—
171C2×Dic6—31no—
172C2×Dic6—91no—
173C3×Q16—181no—
174C3×Q16—181no—
175C3×Q16—181no—
176C3×Q16—181no—
177C3×SD16—181no—
178C3×SD16—181no—
179C3×M16—31no—
180C3×M16—31no—
181C3×M16—31no—
182C3×M16—31no—
183C3×M16—61no—
184C3×M16—61no—
185C3×M16—91no—
186C3×M16—181no—
187C3×M16C2×C3×D311yescharacteristic
188C3×M16C2×C3×D311yescharacteristic
189C2×C3×C8—91no—
190C2×C3×C8—181no—
191#216—61no—
192#216—61no—
193#216—61no—
194#216—181no—
195#216—181no—
196#216—181no—
197#215—61no—
198#215—181no—
199#208—31no—
200#208—91no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—91no—
203Dic12—61no—
204Dic12—181no—
205#204—61no—
206#204—181no—
207ΓL(1,𝔽25)—61no—
208ΓL(1,𝔽25)—181no—
209D3×C8—61no—
210D3×C8—181no—
211C22×C32—21no—
212C22×C32#23511yescharacteristic
213C22×C32C3×#4111yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3×D3—61no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—42no—
219C32×C4—41no—
220C32×C4—41no—
221C32×C4D3×D411yescharacteristic
222C32×C4D3×D411yescharacteristic
223C32×C4C2×C3×D411yescharacteristic
224C32×C4C2×C3×D411yescharacteristic
225C3⋊Dic3—21no—
226C3⋊Dic3—41no—
227C3×Dic3—61no—
228C3×Dic3—121no—
229#137—91no—
230#137—91no—
231C2×Q16—91no—
232#131—91no—
233#108—91no—
234#101—91no—
235#101(C3⋊D3)⋊C311yescharacteristic, complemented
236H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—61no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×Q8—61no—
243C3×Q8—91no—
244C3×Q8—121no—
245C3×Q8—121no—
246C3×Q8—181no—
247C3×Q8—181no—
248C3×D4—91no—
249C3×D4—181no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—62no—
256C2×C3×C4—61no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4C22×C3×D311yescharacteristic
260#67—61no—
261#67—181no—
262C2×Dic3—61no—
263C2×Dic3—181no—
264D12—31no—
265D12—91no—
266C4×D3—61no—
267C4×D3—181no—
268Dic6—31no—
269Dic6—61no—
270Dic6—61no—
271Dic6—91no—
272Dic6—181no—
273Dic6—181no—
274C3×C8—21no—
275C3×C8—21no—
276C3×C8—61no—
277C3×C8—61no—
278C3×C8—61no—
279C3×C8—61no—
280C3×C8—91no—
281C3×C8—91no—
282C3×C8—121no—
283C3×C8—121no—
284C3×C8—181no—
285C3⋊C8—31no—
286C3⋊C8—31no—
287C3⋊C8—61no—
288C3⋊C8—91no—
289C3⋊C8—91no—
290C3⋊C8—181no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—41no—
295C2×C32#88411yescharacteristic
296C2×C32C3×#12111yescharacteristic
297C3⋊D3—41no—
298C3×D3—121no—
299#41—91no—
300C2×Q8—91no—
301C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
302Q16—181no—
303Q16—181no—
304Q16—181no—
305Q16—181no—
306SD16—181no—
307SD16—181no—
308M16—91no—
309M16—181no—
310M16C2×(C3⋊D3)⋊C311yescharacteristic
311M16C2×(C3⋊D3)⋊C311yescharacteristic
312C2×C8—91no—
313C2×C8—181no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—61no—
317C22×C3—181no—
318C22×C3C3×#23511yescharacteristic
319C2×D3—61no—
320C2×D3—181no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—41no—
326C3×C4—61no—
327C3×C4—61no—
328C3×C4—121no—
329C3×C4—121no—
330C3×C4—122no—
331C3×C4—181no—
332C3×C4—361no—
333C3×C4C3×D3×D411yescharacteristic
334C3×C4C3×D3×D411yescharacteristic
335Dic3—61no—
336Dic3—121no—
337Dic3—181no—
338Dic3—361no—
339C32—21no—
340C32#475511yescharacteristic, complemented
341C32C3×#47311yescharacteristic, complemented
342Q8—21no—
343Q8—21no—
344Q8—91no—
345Q8—181no—
346Q8—181no—
347D4—91no—
348D4—181no—
349C2×C4—21no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
353C8—21no—
354C8—21no—
355C8—91no—
356C8—91no—
357C8—181no—
358C2×C3—21no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—61no—
364C2×C3—121no—
365C2×C3—361no—
366C2×C3C3×#88411yescharacteristic
367D3—121no—
368D3—361no—
369C22—181no—
370C22#3828⋊C311yescharacteristic
371C4—41no—
372C4—181no—
373C4—361no—
374C4D4×(C3⋊D3)⋊C311yescharacteristic
375C4D4×(C3⋊D3)⋊C311yescharacteristic
376C3—31no—
377C3—31no—
378C3—61no—
379C3C3×#475511yescharacteristic
380C2—21no—
381C2—361no—
382C2#64093⋊C311yescharacteristic
subgroupG/H
centerC2#64093⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#101C2
socleC2×C3C3×#884
prime radical#3828⋊C3C22
hypercenter#101(C3⋊D3)⋊C3
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
5. H(3,𝔽3)×#101C2semidirect
6. #63651⋊C3C2semidirect
7. #63649⋊C3C2semidirect
8. #10592454C3semidirect
25. C2×#3756C2×C3semidirect
34. C32×#101D3semidirect
35. C32×#101C2×C3semidirect
36. #63651C2×C3semidirect
37. #63649C2×C3semidirect
235. #101(C3⋊D3)⋊C3semidirect
236. H(3,𝔽3)#473semidirect
340. C32#4755semidirect
341. C32C3×#473semidirect

Maximal subgroups

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

subgroupindex
#64151⋊C32
C2×#3756⋊C32
#64118⋊C32
#64116⋊C32
H(3,𝔽3)×#1012
#63651⋊C32
#63649⋊C32
#105924543
C3×#47553