Finite Group Databasebeta

← #10592453⋊C3 #13141701

Subgroups — #10592453⋊C3 #13141701

Order 1728.

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

normal subgroups only

This group has 402 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, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3#64117⋊C3C211yesmaximal, characteristic
4#64116⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
6#63650⋊C3C211yesmaximal, characteristic, complemented
7#63649⋊C3C211yesmaximal, characteristic, complemented
8#10592453C311yesmaximal, characteristic, complemented
9C3×#4754—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14M16×H(3,𝔽3)C2211yescharacteristic
15#3755⋊C3—21no—
16#3755⋊C3—21no—
17#3755⋊C3—21no—
18#3756⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3745⋊C3—21no—
22#3744⋊C3—21no—
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25#64151C2×C311yescharacteristic, complemented
26C2×#3755C2×C311yescharacteristic, complemented
27#64117C2×C311yescharacteristic
28#64116C2×C311yescharacteristic, complemented
29C3×#942—31no—
30C2×C3×#215—31no—
31C3×#908—31no—
32C3×#907—31no—
33C32×#101—21no—
34C32×#101D311yescharacteristic, complemented
35C32×#101C2×C311yescharacteristic, complemented
36#63650C2×C311yescharacteristic, complemented
37#63649C2×C311yescharacteristic, complemented
38C3×#823—31no—
39C3×#822—31no—
40C22×(C3⋊D3)⋊C3—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
49C4×(C3⋊D3)⋊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×#471—91no—
57#4754—31no—
58#4754—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C311yescharacteristic
62#3828C22×C311yescharacteristic
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C3×#235—31no—
65C2×C3×D12—31no—
66C32×M16—21no—
67C32×M16—21no—
68C32×M16C2×D311yescharacteristic
69C32×M16C2×D311yescharacteristic
70C32×M16C22×C311yescharacteristic
71C32×M16C22×C311yescharacteristic
72#3756—21no—
73#3755—21no—
74#3755—21no—
75#3755—21no—
76#3748C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3745—21no—
79#3744—21no—
80#3743—21no—
81C8×C3⋊D3—21no—
82C3×#216—61no—
83C3×#215—61no—
84C3×#215—61no—
85C3×#215—61no—
86C3×#208—31no—
87C2×C3×C3⋊C8—31no—
88C3×D24—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)—41no—
97C4×H(3,𝔽3)C2×D411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99(C3⋊Dic3)⋊C3—21no—
100C3×#137—91no—
101C3×AGL(1,ℤ/8)—91no—
102C2×C3×SD16—91no—
103C3×#131—91no—
104#942—31no—
105#942—91no—
106C2×#215—31no—
107C2×#215—91no—
108#908—31no—
109#908—91no—
110#907—31no—
111#907—91no—
112C3×#108—91no—
113C3×#101—31no—
114C3×#101—31no—
115C3×#101—61no—
116C3×#101C3×D311yescharacteristic
117C3×#100—91no—
118#823—31no—
119#823—91no—
120#822—31no—
121#822—91no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—41no—
129C32×Q8—41no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C23×C311yescharacteristic
136#638—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)C3×D411yescharacteristic
140C4×C3⋊D3—21no—
141#634C3×D411yescharacteristic
142C3×#67—61no—
143C3×D12—31no—
144C3×D12—61no—
145C3×D12—61no—
146C3×C4×D3—61no—
147C3×Dic6—31no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—21no—
152C32×C8—41no—
153C32×C8—41no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)C3×D411yescharacteristic
156C3⋊(C3⋊C8)C3×D411yescharacteristic
157C3×C3⋊C8—31no—
158C3×C3⋊C8—31no—
159C3×C3⋊C8—61no—
160#471—91noSylow
161C2×H(3,𝔽3)—21no—
162C2×H(3,𝔽3)#12111yescharacteristic
163(C3⋊D3)⋊C3—41no—
164(C3⋊D3)⋊C3—81no—
165C3×#41—91no—
166C2×C3×Q8—31no—
167C2×C3×Q8—31no—
168C2×C3×Q8—61no—
169C2×C3×Q8C2×C3×D311yescharacteristic
170C2×C3×D4—91no—
171#235—31no—
172#235—91no—
173C2×D12—31no—
174C2×D12—91no—
175C3×Q16—181no—
176C3×SD16—181no—
177C3×SD16—181no—
178C3×SD16—181no—
179C3×SD16—181no—
180C3×D8—181no—
181C3×M16—31no—
182C3×M16—31no—
183C3×M16—31no—
184C3×M16—31no—
185C3×M16—61no—
186C3×M16—61no—
187C3×M16—91no—
188C3×M16—181no—
189C3×M16C2×C3×D311yescharacteristic
190C3×M16C2×C3×D311yescharacteristic
191C2×C3×C8—91no—
192C2×C3×C8—181no—
193#216—61no—
194#216—181no—
195#215—61no—
196#215—61no—
197#215—61no—
198#215—181no—
199#215—181no—
200#215—181no—
201#208—31no—
202#208—91no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205D24—61no—
206D24—181no—
207#204—61no—
208#204—181no—
209ΓL(1,𝔽25)—61no—
210ΓL(1,𝔽25)—181no—
211D3×C8—61no—
212D3×C8—181no—
213C22×C32—21no—
214C22×C32#23511yescharacteristic
215C22×C32C3×#4111yescharacteristic
216C2×C3⋊D3—21no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—81no—
219C2×C3×D3—61no—
220C2×C3×D3—121no—
221C2×C3×D3—241no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—42no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4D3×D411yescharacteristic
228C32×C4D3×D411yescharacteristic
229C32×C4C2×C3×D411yescharacteristic
230C32×C4C2×C3×D411yescharacteristic
231C3⋊Dic3—21no—
232C3×Dic3—61no—
233#137—91no—
234AGL(1,ℤ/8)—91no—
235C2×SD16—91no—
236#131—91no—
237#108—91no—
238#101(C3⋊D3)⋊C311yescharacteristic, complemented
239#100—91no—
240H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
241C23×C3—181no—
242C22×D3—61no—
243C22×D3—181no—
244C3×Q8—21no—
245C3×Q8—21no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—61no—
249C3×Q8—61no—
250C3×Q8—91no—
251C3×Q8—121no—
252C3×Q8—121no—
253C3×D4—91no—
254C3×D4—181no—
255C3×D4—181no—
256C3×D4—181no—
257C2×C3×C4—21no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—62no—
263C2×C3×C4—61no—
264C2×C3×C4—181no—
265C2×C3×C4C22×C3×D311yescharacteristic
266#67—61no—
267#67—181no—
268D12—31no—
269D12—61no—
270D12—61no—
271D12—91no—
272D12—181no—
273D12—181no—
274C4×D3—61no—
275C4×D3—181no—
276Dic6—31no—
277Dic6—91no—
278C3×C8—21no—
279C3×C8—21no—
280C3×C8—61no—
281C3×C8—61no—
282C3×C8—61no—
283C3×C8—61no—
284C3×C8—91no—
285C3×C8—91no—
286C3×C8—121no—
287C3×C8—121no—
288C3×C8—181no—
289C3⋊C8—31no—
290C3⋊C8—31no—
291C3⋊C8—61no—
292C3⋊C8—91no—
293C3⋊C8—91no—
294C3⋊C8—181no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—41no—
299C2×C32#88411yescharacteristic
300C2×C32C3×#12111yescharacteristic
301C3⋊D3—41no—
302C3⋊D3—81no—
303C3×D3—121no—
304C3×D3—241no—
305#41—91no—
306C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
307C2×D4—91no—
308Q16—181no—
309SD16—181no—
310SD16—181no—
311SD16—181no—
312SD16—181no—
313D8—181no—
314M16—91no—
315M16—181no—
316M16C2×(C3⋊D3)⋊C311yescharacteristic
317M16C2×(C3⋊D3)⋊C311yescharacteristic
318C2×C8—91no—
319C2×C8—181no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—61no—
323C22×C3—181no—
324C22×C3—361no—
325C22×C3—721no—
326C22×C3C3×#23511yescharacteristic
327C2×D3—61no—
328C2×D3—121no—
329C2×D3—181no—
330C2×D3—241no—
331C2×D3—361no—
332C2×D3—721no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—41no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—121no—
341C3×C4—121no—
342C3×C4—122no—
343C3×C4—181no—
344C3×C4C3×D3×D411yescharacteristic
345C3×C4C3×D3×D411yescharacteristic
346Dic3—61no—
347Dic3—181no—
348C32—21no—
349C32#475411yescharacteristic, complemented
350C32C3×#47111yescharacteristic, complemented
351C23—181no—
352Q8—21no—
353Q8—21no—
354Q8—91no—
355D4—91no—
356D4—181no—
357D4—181no—
358D4—181no—
359C2×C4—21no—
360C2×C4—181no—
361C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
362C8—21no—
363C8—21no—
364C8—91no—
365C8—91no—
366C8—181no—
367C2×C3—21no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—121no—
374C2×C3—361no—
375C2×C3—721no—
376C2×C3C3×#88411yescharacteristic
377D3—121no—
378D3—241no—
379D3—361no—
380D3—721no—
381C22—181no—
382C22—361no—
383C22—721no—
384C22#3828⋊C311yescharacteristic
385C4—41no—
386C4—181no—
387C4D4×(C3⋊D3)⋊C311yescharacteristic
388C4D4×(C3⋊D3)⋊C311yescharacteristic
389C3—31no—
390C3—31no—
391C3—61no—
392C3C3×#475411yescharacteristic
393C2—21no—
394C2—361no—
395C2—721no—
396C2#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#64117⋊C3C2
hypercenter#101(C3⋊D3)⋊C3
hypocenterC32C3×#471
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
1. #64151⋊C3C2semidirect
2. C2×#3755⋊C3C2semidirect
4. #64116⋊C3C2semidirect
5. H(3,𝔽3)×#101C2semidirect
6. #63650⋊C3C2semidirect
7. #63649⋊C3C2semidirect
8. #10592453C3semidirect
25. #64151C2×C3semidirect
26. C2×#3755C2×C3semidirect
28. #64116C2×C3semidirect
34. C32×#101D3semidirect
35. C32×#101C2×C3semidirect
36. #63650C2×C3semidirect
37. #63649C2×C3semidirect
238. #101(C3⋊D3)⋊C3semidirect
240. H(3,𝔽3)#471semidirect
349. C32#4754semidirect
350. C32C3×#471semidirect

Maximal subgroups

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

subgroupindex
#64151⋊C32
C2×#3755⋊C32
#64117⋊C32
#64116⋊C32
H(3,𝔽3)×#1012
#63650⋊C32
#63649⋊C32
#105924533
C3×#47543