Finite Group Databasebeta

← #13159435

Subgroups — #13159435

Order 1728.

1140 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 178 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
1C2×#89014C211yesmaximal, characteristic
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#88787C211yesmaximal, characteristic, complemented
4#11722180C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592470—31nomaximal
9#10590607—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×D3×C3⋊Dic3—21no—
16#89014—22no—
17#89014C414yescomplemented
18C33×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88787—22no—
24C2×#3830—31no—
25C2×#3795—34no—
26C2×#3743—31no—
27C4⋊C4×C3⋊D3D311yescharacteristic, complemented
28C3×D3×C4⋊C4—34no—
29C2×#3711—34no—
30C32×#102—31no—
31#63659—31no—
32#63643—31no—
33#63637—31no—
34#63637—34no—
35#63637—64no—
36C3×#809D314yescomplemented
37#63567—34no—
38#63562—34no—
39C22×C3×C3⋊D3D411yescharacteristic
40C22×C32×D3—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308C2×C412yescomplemented
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—42no—
49C3×C4×C3⋊D3C2×C412yes—
50C3×C4×C3⋊D3D412yes—
51C32×D12—21no—
52C32×D12C2×C412yescomplemented
53#6290—24no—
54D3×C3⋊Dic3—24no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C32×C3⋊C8—21no—
57#4966—91no—
58#4771—94no—
59C2×C32×D4—31no—
60C2×#638—61no—
61C22×C3⋊Dic3—61no—
62#3830—34no—
63#3830—62no—
64C2×C4×C3⋊D3—21no—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×#634—31no—
67C2×#634—34no—
68C2×#634—64no—
69C2×C3×D12C2×D314yes—
70C2×C3×C4×D3—34no—
71C2×C3×C4×D3—64no—
72C2×#625—64no—
73C2×D3×Dic3—64no—
74#3795—316no—
75#3795—68no—
76C2×C32×C8—31no—
77C32×C4⋊C4—24no—
78C32×C4⋊C4C2×D314yes—
79C32×C4⋊C4C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—21no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85#3743—62no—
86C3×C4⋊Dic3—34no—
87C3×Dic3⋊C4—64no—
88C2×C3×C3⋊C8C2×D314yes—
89#3711—68no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C3×C3⋊D3C22⋊C412yes—
94C2×C32×D3—22no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4—41no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)—22no—
101C3×C3⋊Dic3—41no—
102C3×C3⋊Dic3C22⋊C412yes—
103C2×#237—94no—
104C2×#235—91no—
105C2×C3×C4⋊C4—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—94no—
108D3×C4⋊C4—34no—
109C3×#102—34no—
110#832—94no—
111#816—94no—
112#810—94no—
113#810—91no—
114#810—91no—
115#810—184no—
116#809—91no—
117#809C3⋊D311yescharacteristic, complemented
118C23×C32—61no—
119C22×C3⋊D3#6711yescharacteristic
120C22×C3×D3—24no—
121C22×C3×D3—34no—
122C32×D4—32no—
123C32×D4—61no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—44no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—64no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—124no—
136C2×C3⋊Dic3#6711yescharacteristic
137C4×C3⋊D3—42no—
138C4×C3⋊D3C4×D312yes—
139C4×C3⋊D3D1212yes—
140#634—32no—
141#634—38no—
142#634—68no—
143#634—61no—
144#634—64no—
145#634—124no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—64no—
148C3×D12—24no—
149C3×D12C4×D318yescomplemented
150C3×C4×D3—38no—
151C3×C4×D3—38no—
152C3×C4×D3—128no—
153#625—616no—
154D3×Dic3—616no—
155C32×C8—61no—
156C3⋊(C3⋊C8)—61no—
157C3⋊(C3⋊C8)—64no—
158C3⋊(C3⋊C8)—124no—
159C3×C3⋊C8—24no—
160#417—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C2×C33#13711yescharacteristic
164C3×C3⋊D3—22no—
165C32×D3—42no—
166C2×C3×D4—34no—
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169C2×#67—181no—
170C2×#67—184no—
171C22×Dic3—184no—
172#237—916no—
173#237—188no—
174#235—94no—
175#235—182no—
176C2×D12C2×C3⋊D311yescharacteristic
177C2×C4×D3—34no—
178C2×C4×D3—64no—
179C2×C4×D3—181no—
180C2×Dic6—94no—
181C2×Dic6—91no—
182C2×Dic6—184no—
183C2×C3×C8—34no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—91no—
186C3×C4⋊C4—181no—
187C3×C4⋊C4D3214yes—
188C3×C4⋊C4C2×C3⋊D311yescharacteristic
189C4⋊Dic3—34no—
190Dic3⋊C4—64no—
191#208—182no—
192C2×C3⋊C8—94no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—184no—
195C2×C3⋊C8C2×C3⋊D311yescharacteristic
196ΓL(1,𝔽25)—188no—
197C22×C32—24no—
198C22×C32—62no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32D3×D414yes—
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3(C2×D3)⋊C412yes—
206C2×C3×D3—28no—
207C2×C3×D3—38no—
208C2×C3×D3—44no—
209C2×C3×D3—44no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—41no—
215C32×C4—44no—
216C32×C4—84no—
217C32×C4C2×C4×D314yes—
218C32×C4C2×C4×D311yescharacteristic
219C32×C4C2×D1211yescharacteristic
220C32×C4D3×D414yes—
221C3⋊Dic3—41no—
222C3⋊Dic3—62no—
223C3⋊Dic3—68no—
224C3⋊Dic3—128no—
225C3⋊Dic3(C2×D3)⋊C412yes—
226C3×Dic3—38no—
227C3×Dic3—124no—
228C2×#41—271no—
229C2×M16—271no—
230C2×C4⋊C4—91no—
231#103—271no—
232#103—271no—
233#102—31no—
234#102—271no—
235C33#41711yescharacteristic, complemented, Sylow
236C23×C3—64no—
237C23×C3—91no—
238C22×D3—21no—
239C22×D3—34no—
240C3×D4—38no—
241C3×D4—64no—
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—44no—
246C2×C3×C4—91no—
247C2×C3×C4—92no—
248C2×C3×C4—92no—
249C2×C3×C4—181no—
250C2×C3×C4—362no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—184no—
254#67—1816no—
255C2×Dic3—34no—
256C2×Dic3—64no—
257C2×Dic3—1816no—
258C2×Dic3—184no—
259C2×Dic3—181no—
260C2×Dic3—364no—
261D12—21no—
262D12C4×C3⋊D312yescomplemented
263C4×D3—38no—
264C4×D3—38no—
265C4×D3—128no—
266C4×D3—184no—
267Dic6—98no—
268Dic6—92no—
269Dic6—188no—
270Dic6—184no—
271Dic6—181no—
272Dic6—364no—
273C3×C8—64no—
274C3⋊C8—21no—
275C3⋊C8—184no—
276C3⋊C8—181no—
277C3⋊C8—364no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32—122no—
282C2×C32D3×C22⋊C414yes—
283C2×C32#91114yes—
284C2×C32#91514yes—
285C2×C32C2×(C2×D3)⋊C411yescharacteristic
286C2×C32#93211yescharacteristic
287C2×C32#94211yescharacteristic
288C3⋊D3—22no—
289C3×D3—48no—
290C3×D3—68no—
291#41—274no—
292#41—542no—
293C2×Q8—271no—
294C2×D4—31no—
295C2×D4—541no—
296C22×C4—91no—
297C22×C4—181no—
298C22×C4—541no—
299M16—542no—
300C2×C8—31no—
301C2×C8—271no—
302C4⋊C4—91no—
303C4⋊C4—181no—
304C4⋊C4D3×C3⋊D311yescharacteristic
305C22×C3—24no—
306C22×C3—68no—
307C22×C3—92no—
308C22×C3—124no—
309C22×C3—124no—
310C22×C3—181no—
311C22×C3—181no—
312C22×C3D3×#6714yes—
313C22×C3D4×C3⋊D311yescharacteristic
314C2×D3—22no—
315C2×D3—38no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—64no—
319C2×D3—64no—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4—41no—
323C3×C4—44no—
324C3×C4—84no—
325C3×C4—92no—
326C3×C4—361no—
327C3×C4C4×D3214yes—
328C3×C4D3×D1214yes—
329C3×C4C2×C4×C3⋊D311yescharacteristic
330C3×C4D4×C3⋊D311yescharacteristic
331Dic3—38no—
332Dic3—124no—
333Dic3—182no—
334Dic3—188no—
335Dic3—368no—
336C32—24no—
337C32#477114yescomplemented
338C32#496611yescharacteristic, complemented
339C23—61no—
340C23—91no—
341Q8—272no—
342Q8—541no—
343D4—32no—
344D4—61no—
345D4—544no—
346C2×C4—21no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—181no—
351C2×C4—362no—
352C2×C4—544no—
353C2×C4—541no—
354C2×C4C2×D3×C3⋊D311yescharacteristic
355C8—61no—
356C8—541no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—128no—
361C2×C3—182no—
362C2×C3D3×(C2×D3)⋊C414yes—
363C2×C3#6393214yes—
364C2×C3#6394614yes—
365C2×C3C22⋊C4×C3⋊D311yescharacteristic
366C2×C3#6412011yescharacteristic
367C2×C3#6412411yescharacteristic
368D3—42no—
369D3—68no—
370C22—62no—
371C22—92no—
372C22—121no—
373C22—121no—
374C22—181no—
375C22—181no—
376C22C3⋊D3×#6711yescharacteristic
377C4—41no—
378C4—92no—
379C4—361no—
380C4—542no—
381C4C4×D3×C3⋊D311yescharacteristic
382C4C3⋊D3×D1211yescharacteristic
383C3—24no—
384C3#1059060714yescomplemented
385C3#1059247011yescharacteristic, complemented
386C2—122no—
387C2—182no—
388C2C3⋊D3×(C2×D3)⋊C411yescharacteristic
389C2#1172356011yescharacteristic
390C2#1172358411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89014C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊C4×C3⋊D3C2semidirect
3. C2×#88787C2semidirect
4. #11722180C2semidirect
5. C32×#809C2semidirect
6. #11722042C2semidirect
7. #11722031C2semidirect
17. #89014C4semidirect
27. C4⋊C4×C3⋊D3D3semidirect
36. C3×#809D3semidirect
45. #6308C2×C4semidirect
52. C32×D12C2×C4semidirect
117. #809C3⋊D3semidirect
149. C3×D12C4×D3semidirect
235. C33#417semidirect
262. D12C4×C3⋊D3semidirect
337. C32#4771semidirect
338. C32#4966semidirect
384. C3#10590607semidirect
385. C3#10592470semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C3×C4⋊C4×C3⋊D32
C2×#887872
#117221802
C32×#8092
#117220422
#117220312
#105924703
#105906073