Finite Group Databasebeta

← #10592597⋊C3 #13142271

Subgroups — #10592597⋊C3 #13142271

Order 1728.

394 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 86 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#64118⋊C3C211yesmaximal, characteristic, complemented
2#64117⋊C3C211yesmaximal, characteristic
3C2×#3744⋊C3C211yesmaximal, characteristic, complemented
4#64110⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#108C211yesmaximal, characteristic, complemented
6#63650⋊C3C211yesmaximal, characteristic, complemented
7#63651⋊C3C211yesmaximal, characteristic, complemented
8#10592597C311yesmaximal, characteristic, complemented
9C3×#4898—31nomaximal
10#3828⋊C3C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14M16×H(3,𝔽3)C2211yescharacteristic
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3748⋊C3C2211yescharacteristic
17#3745⋊C3—21no—
18#3744⋊C3—21no—
19#3744⋊C3—21no—
20#3744⋊C3—21no—
21#3744⋊C3—21no—
22#3743⋊C3—21no—
23C8×(C3⋊D3)⋊C3—21no—
24#3746⋊C3—21no—
25#64118C2×C311yescharacteristic, complemented
26#64117C2×C311yescharacteristic
27C2×#3744C2×C311yescharacteristic, complemented
28#64110C2×C311yescharacteristic, complemented
29C3×#909—31no—
30C3×#908—31no—
31C2×C3×#204—31no—
32C3×#901—31no—
33C32×#108—21no—
34C32×#108D311yescharacteristic, complemented
35C32×#108C2×C311yescharacteristic, complemented
36#63651C2×C311yescharacteristic, complemented
37#63650C2×C311yescharacteristic, complemented
38C3×#824—31no—
39C3×#823—31no—
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#638⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊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—
53C8×H(3,𝔽3)D411yescharacteristic
54C8×H(3,𝔽3)D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3—21no—
56C3×#471—91no—
57#4898—31no—
58#4898—91no—
59#3828C22×C311yescharacteristic
60C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic
62C3×#235—31no—
63C2×C3×D12—31no—
64C2×C3×Dic6—31no—
65C32×M16—21no—
66C32×M16—21no—
67C32×M16C2×D311yescharacteristic
68C32×M16C2×D311yescharacteristic
69C32×M16C22×C311yescharacteristic
70C32×M16C22×C311yescharacteristic
71C2×C32×C8—21no—
72C2×C32×C8C2×D311yescharacteristic
73C2×C32×C8C22×C311yescharacteristic
74#3748C22×C311yescharacteristic
75#3746—21no—
76#3745—21no—
77#3744—21no—
78#3744—21no—
79#3744—21no—
80#3744—21no—
81#3743—21no—
82C8×C3⋊D3—21no—
83C3×#208—31no—
84C3×Dic12—61no—
85C3×D24—61no—
86C3×#204—61no—
87C3×#204—61no—
88C3×#204—61no—
89C3×#204—61no—
90C3×ΓL(1,𝔽25)—61no—
91C3×D3×C8—61no—
92C22×H(3,𝔽3)#4111yescharacteristic
93C2×(C3⋊D3)⋊C3—21no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—81no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100C3×#137—91no—
101C3×AGL(1,ℤ/8)—91no—
102C2×C3×SD16—91no—
103C3×#131—91no—
104#909—31no—
105#909—91no—
106#908—31no—
107#908—91no—
108C2×#204—31no—
109C2×#204—91no—
110#901—31no—
111#901—91no—
112C3×#108—31no—
113C3×#108—31no—
114C3×#108—61no—
115C3×#108C3×D311yescharacteristic
116C3×#101—91no—
117C3×#100—91no—
118#824—31no—
119#824—91no—
120#823—31no—
121#823—91no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C23×C311yescharacteristic
127#638—21no—
128C2×C3⋊Dic3—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)C3×D411yescharacteristic
132C4×C3⋊D3—21no—
133#634—21no—
134#634—21no—
135#634C3×D411yescharacteristic
136C3×#67—61no—
137C2×C3×Dic3—61no—
138C3×D12—31no—
139C3×D12—61no—
140C3×D12—61no—
141C3×C4×D3—61no—
142C3×Dic6—31no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—21no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—41no—
152C32×C8—41no—
153C32×C8D1211yescharacteristic
154C32×C8D1211yescharacteristic
155C32×C8C3×D411yescharacteristic
156C32×C8C3×D411yescharacteristic
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—61no—
159#471—91noSylow
160C2×H(3,𝔽3)—21no—
161C2×H(3,𝔽3)#12111yescharacteristic
162(C3⋊D3)⋊C3—41no—
163(C3⋊D3)⋊C3—81no—
164C3×#41—91no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167#235—31no—
168#235—91no—
169C2×D12—31no—
170C2×D12—91no—
171C2×Dic6—31no—
172C2×Dic6—91no—
173C3×Q16—181no—
174C3×SD16—181no—
175C3×SD16—181no—
176C3×SD16—181no—
177C3×SD16—181no—
178C3×D8—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—31no—
190C2×C3×C8—31no—
191C2×C3×C8—61no—
192C2×C3×C8—181no—
193C2×C3×C8C2×C3×D311yescharacteristic
194#208—31no—
195#208—91no—
196Dic12—61no—
197Dic12—181no—
198D24—61no—
199D24—181no—
200#204—61no—
201#204—61no—
202#204—61no—
203#204—61no—
204#204—181no—
205#204—181no—
206#204—181no—
207#204—181no—
208ΓL(1,𝔽25)—61no—
209ΓL(1,𝔽25)—181no—
210D3×C8—61no—
211D3×C8—181no—
212C22×C32—21no—
213C22×C32#23911yescharacteristic
214C22×C32C3×#4111yescharacteristic
215C2×C3⋊D3—21no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—81no—
218C2×C3×D3—61no—
219C2×C3×D3—121no—
220C2×C3×D3—241no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4C2×D1211yescharacteristic
224C32×C4D3×D411yescharacteristic
225C32×C4C2×C3×D411yescharacteristic
226C32×C4C2×C3×D411yescharacteristic
227C3⋊Dic3—21no—
228C3⋊Dic3—41no—
229C3×Dic3—61no—
230C3×Dic3—121no—
231#137—91no—
232AGL(1,ℤ/8)—91no—
233C2×SD16—91no—
234#131—91no—
235#108(C3⋊D3)⋊C311yescharacteristic, complemented
236#101—91no—
237#100—91no—
238H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
239C23×C3—181no—
240C22×D3—61no—
241C22×D3—181no—
242C3×Q8—91no—
243C3×Q8—181no—
244C3×Q8—181no—
245C3×D4—91no—
246C3×D4—181no—
247C3×D4—181no—
248C3×D4—181no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—61no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4C22×C3×D311yescharacteristic
255#67—61no—
256#67—181no—
257C2×Dic3—61no—
258C2×Dic3—181no—
259D12—31no—
260D12—61no—
261D12—61no—
262D12—91no—
263D12—181no—
264D12—181no—
265C4×D3—61no—
266C4×D3—181no—
267Dic6—31no—
268Dic6—61no—
269Dic6—61no—
270Dic6—91no—
271Dic6—181no—
272Dic6—181no—
273C3×C8—21no—
274C3×C8—21no—
275C3×C8—31no—
276C3×C8—31no—
277C3×C8—31no—
278C3×C8—31no—
279C3×C8—61no—
280C3×C8—61no—
281C3×C8—61no—
282C3×C8—61no—
283C3×C8—61no—
284C3×C8—61no—
285C3×C8—121no—
286C3×C8—121no—
287C3×C8—181no—
288C3×C8C3×D1211yescharacteristic
289C3×C8C3×D1211yescharacteristic
290C3⋊C8—61no—
291C3⋊C8—181no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—41no—
296C2×C32#89511yescharacteristic
297C2×C32C3×#12111yescharacteristic
298C3⋊D3—41no—
299C3⋊D3—81no—
300C3×D3—121no—
301C3×D3—241no—
302#41—91no—
303C2×Q8—91no—
304C2×D4—91no—
305Q16—181no—
306SD16—181no—
307SD16—181no—
308SD16—181no—
309SD16—181no—
310D8—181no—
311M16—91no—
312M16—181no—
313M16C2×(C3⋊D3)⋊C311yescharacteristic
314M16C2×(C3⋊D3)⋊C311yescharacteristic
315C2×C8—181no—
316C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—61no—
320C22×C3—181no—
321C22×C3—361no—
322C22×C3—721no—
323C22×C3C3×#23911yescharacteristic
324C2×D3—61no—
325C2×D3—121no—
326C2×D3—181no—
327C2×D3—241no—
328C2×D3—361no—
329C2×D3—721no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—61no—
335C3×C4—61no—
336C3×C4—181no—
337C3×C4—361no—
338C3×C4C2×C3×D1211yescharacteristic
339C3×C4C3×D3×D411yescharacteristic
340Dic3—61no—
341Dic3—121no—
342Dic3—181no—
343Dic3—361no—
344C32—21no—
345C32#489811yescharacteristic, complemented
346C32C3×#47111yescharacteristic, complemented
347C23—181no—
348Q8—91no—
349Q8—181no—
350Q8—181no—
351D4—91no—
352D4—181no—
353D4—181no—
354D4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
358C8—21no—
359C8—21no—
360C8—181no—
361C8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
362C8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
363C2×C3—21no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—61no—
369C2×C3—121no—
370C2×C3—361no—
371C2×C3—721no—
372C2×C3C3×#89511yescharacteristic
373D3—121no—
374D3—241no—
375D3—361no—
376D3—721no—
377C22—181no—
378C22—361no—
379C22—721no—
380C22#8872211yescharacteristic
381C4—181no—
382C4—361no—
383C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
384C4D4×(C3⋊D3)⋊C311yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3C3×#489811yescharacteristic
389C2—21no—
390C2—361no—
391C2—721no—
392C2#64104⋊C311yescharacteristic
subgroupG/H
centerC2#64104⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#108C2
socleC2×C3C3×#895
prime radical#64117⋊C3C2
hypercenter#108(C3⋊D3)⋊C3
hypocenterC32C3×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
1. #64118⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
4. #64110⋊C3C2semidirect
5. H(3,𝔽3)×#108C2semidirect
6. #63650⋊C3C2semidirect
7. #63651⋊C3C2semidirect
8. #10592597C3semidirect
25. #64118C2×C3semidirect
27. C2×#3744C2×C3semidirect
28. #64110C2×C3semidirect
34. C32×#108D3semidirect
35. C32×#108C2×C3semidirect
36. #63651C2×C3semidirect
37. #63650C2×C3semidirect
235. #108(C3⋊D3)⋊C3semidirect
238. H(3,𝔽3)#471semidirect
345. C32#4898semidirect
346. C32C3×#471semidirect

Maximal subgroups

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

subgroupindex
#64118⋊C32
#64117⋊C32
C2×#3744⋊C32
#64110⋊C32
H(3,𝔽3)×#1082
#63650⋊C32
#63651⋊C32
#105925973
C3×#48983