Finite Group Databasebeta

← #13137856

Subgroups — #13137856

Order 1728.

578 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 101 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×#88634C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C2×#88414C211yesmaximal, characteristic, complemented
4C9×#810C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720283C211yesmaximal, characteristic, complemented
8#10590612—31nomaximal
9#10588483—31nomaximal
10C2×C4×C3⋊D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×#6192—21no—
15C2×Dic3×D9—21no—
16#88634—22no—
17#88634C414yescomplemented
18C3×C9×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20#88534—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88414—22no—
24C2×#3796—31no—
25C4⋊C4×C3⋊D3—31no—
26C2×#3712—31no—
27C2×#3701—31no—
28C3×#810D311yescharacteristic
29C3×#809—31no—
30#63566—31no—
31#63563—31no—
32C2×#3663—31no—
33C4⋊C4×D9—31no—
34C4⋊C4×D9—61no—
35C9×#103—31no—
36#63395—31no—
37#63380—31no—
38#63369D311yescharacteristic, complemented
39C22×C3⋊D9D411yescharacteristic
40C22×C3×D9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C4×C3⋊D9—42no—
46C4×C3⋊D9C2×C412yes—
47C4×C3⋊D9D412yes—
48C2×C9×Dic3—21no—
49C3×D36—21no—
50C3×D36C2×C412yescomplemented
51C9×Dic6—21no—
52C9×Dic6C2×C412yescomplemented
53#6192—24no—
54Dic3×D9—24no—
55C9×C3⋊C8—21no—
56C3×C9⋊C8—21no—
57#4803—91no—
58#4771—91no—
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—61no—
61C2×C3×D12—31no—
62C2×C3×Dic6C2×D311yescharacteristic
63C2×#626—61no—
64C2×D3×Dic3—61no—
65#3796—34no—
66#3796—62no—
67C32×C4⋊C4C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)—31no—
69#3750—61no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8C2×D311yescharacteristic
72#3712—62no—
73C2×Q8×C9—31no—
74#3701—34no—
75#3701—62no—
76C2×D36—61no—
77C2×D36C2×D311yescharacteristic
78C2×C4×D9—31no—
79C2×C4×D9—61no—
80C2×C4×D9—61no—
81C2×C4×D9—61no—
82C2×C4×D9—62no—
83C2×C8×C9—31no—
84C9×C4⋊C4—21no—
85C9×C4⋊C4C2×D311yescharacteristic
86C4⋊Dic9—31no—
87C4⋊Dic9—61no—
88Dic9⋊C4—61no—
89Dic9⋊C4—62no—
90C2×C9⋊C8C2×D311yescharacteristic
91#3663—62no—
92C22×C3×C9C2×D411yescharacteristic
93C2×C3⋊D9—21no—
94C2×C3⋊D9—21no—
95C2×C3⋊D9C22⋊C412yes—
96C2×C3×D9—22no—
97C2×C3×D9—41no—
98C2×C3×D9—41no—
99C3×C4×C9—41no—
100C3×C4×C9C22×C411yescharacteristic
101C3×C4×C9C2×D411yescharacteristic
102C3⋊Dic9—41no—
103C3⋊Dic9C22⋊C412yes—
104C9×Dic3—22no—
105C2×#239—91no—
106C2×#237—91no—
107C2×ΓL(1,𝔽25)—91no—
108C2×ΓL(1,𝔽25)—91no—
109D3×C4⋊C4—91no—
110D3×C4⋊C4—91no—
111D3×C4⋊C4—181no—
112C3×#103—31no—
113C3×#102—91no—
114#835—91no—
115#832—91no—
116#821—91no—
117#816—91no—
118#810D911yescharacteristic, complemented
119#809—31no—
120C22×C3⋊D3—31no—
121C22×C3×D3—61no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—61no—
126C4×C3⋊D3—32no—
127C4×C3⋊D3—32no—
128C4×C3⋊D3—122no—
129C2×C3×Dic3—21no—
130C3×D12—32no—
131C3×D12—61no—
132C3×Dic6—21no—
133C3×Dic6C4×D312yes—
134#626—64no—
135D3×Dic3—64no—
136C22×D9—21no—
137C22×D9—31no—
138C22×D9—61no—
139C3×C3⋊C8—21no—
140C3×C3⋊C8—61no—
141Q8×C9—32no—
142Q8×C9—61no—
143C2×C4×C9—22no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—61no—
147C2×C4×C9C22×D311yescharacteristic
148C2×Dic9—31no—
149C2×Dic9—61no—
150C2×Dic9—62no—
151C2×Dic9—61no—
152D36—21no—
153D36—64no—
154D36C4×D312yescomplemented
155C4×D9—32no—
156C4×D9—32no—
157C4×D9—62no—
158C4×D9—62no—
159C4×D9—64no—
160C4×D9—124no—
161C4×D9—122no—
162C8×C9—61no—
163C9⋊C8—21no—
164#417—271noSylow
165C2×C3×C9C2×C22⋊C411yescharacteristic
166C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
167C2×C3×C9#13711yescharacteristic
168C3⋊D9—22no—
169C3×D9—42no—
170C2×C3×Q8—31no—
171C2×C3×D4—91no—
172C2×#67—181no—
173C22×Dic3—181no—
174#239—94no—
175#239—182no—
176#237—94no—
177#237—182no—
178C2×D12—31no—
179C2×D12—181no—
180C2×C4×D3—91no—
181C2×C4×D3—91no—
182C2×C4×D3—181no—
183C2×C4×D3—181no—
184C2×C4×D3—181no—
185C2×C4×D3—181no—
186C2×C4×D3—182no—
187C2×Dic6C2×D911yescharacteristic
188C2×C3×C8—31no—
189C2×C3×C8—91no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4C2×D911yescharacteristic
192C3×C4⋊C4D3211yescharacteristic
193C4⋊Dic3—91no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196Dic3⋊C4—181no—
197Dic3⋊C4—181no—
198Dic3⋊C4—182no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8C2×D911yescharacteristic
201ΓL(1,𝔽25)—182no—
202ΓL(1,𝔽25)—182no—
203C22×C32D3×D411yescharacteristic
204C2×C3⋊D3—32no—
205C2×C3⋊D3—61no—
206C2×C3⋊D3—61no—
207C2×C3×D3—62no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—41no—
211C32×C4C2×C4×D311yescharacteristic
212C32×C4D3×D411yescharacteristic
213C3⋊Dic3—32no—
214C3⋊Dic3—121no—
215C3×Dic3—22no—
216C22×C9—21no—
217C22×C9D3×D411yescharacteristic
218C2×D9—22no—
219C2×D9—32no—
220C2×D9—41no—
221C2×D9—41no—
222C2×D9—62no—
223C2×D9—62no—
224C2×D9—62no—
225C2×D9—61no—
226C2×D9—61no—
227C4×C9—21no—
228C4×C9—21no—
229C4×C9—42no—
230C4×C9—41no—
231C4×C9—62no—
232C4×C9C2×C4×D311yescharacteristic
233C4×C9D3×D411yescharacteristic
234Dic9—32no—
235Dic9—62no—
236Dic9—122no—
237Dic9—121no—
238C2×#41—271no—
239C2×M16—271no—
240C2×C4⋊C4—271no—
241#103—31no—
242#103—271no—
243#102—91no—
244#102—271no—
245C3×C9#41711yescharacteristic, complemented, Sylow
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—91no—
249C22×D3—91no—
250C22×D3—181no—
251C3×Q8—32no—
252C3×Q8—61no—
253C3×D4—92no—
254C3×D4—181no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—61no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262#67—184no—
263C2×Dic3—21no—
264C2×Dic3—91no—
265C2×Dic3—91no—
266C2×Dic3—181no—
267C2×Dic3—184no—
268C2×Dic3—182no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271D12—32no—
272D12—61no—
273D12—184no—
274C4×D3—92no—
275C4×D3—92no—
276C4×D3—92no—
277C4×D3—92no—
278C4×D3—182no—
279C4×D3—182no—
280C4×D3—184no—
281C4×D3—362no—
282C4×D3—364no—
283C4×D3—362no—
284Dic6—21no—
285Dic6C4×D912yescomplemented
286C3×C8—61no—
287C3×C8—181no—
288C3⋊C8—21no—
289C3⋊C8—61no—
290C2×C32D3×C22⋊C411yescharacteristic
291C2×C32#91411yescharacteristic
292C2×C32#91811yescharacteristic
293C3⋊D3—62no—
294C3×D3—122no—
295C2×C9—21no—
296C2×C9—21no—
297C2×C9—21no—
298C2×C9D3×C22⋊C411yescharacteristic
299C2×C9#91111yescharacteristic
300C2×C9#91511yescharacteristic
301D9—42no—
302D9—64no—
303D9—62no—
304#41—274no—
305#41—542no—
306C2×Q8—31no—
307C2×D4—91no—
308C2×D4—541no—
309C22×C4—271no—
310C22×C4—541no—
311C22×C4—541no—
312M16—542no—
313C2×C8—31no—
314C2×C8—91no—
315C4⋊C4—271no—
316C4⋊C4—541no—
317C4⋊C4D3×D911yescharacteristic
318C22×C3—21no—
319C22×C3—182no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3D4×D911yescharacteristic
323C22×C3#381111yescharacteristic
324C2×D3—62no—
325C2×D3—92no—
326C2×D3—92no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—182no—
330C2×D3—182no—
331C2×D3—182no—
332C2×D3—181no—
333C2×D3—181no—
334C2×D3—181no—
335C2×D3—181no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—42no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—62no—
342C3×C4C2×C4×D911yescharacteristic
343C3×C4D4×D911yescharacteristic
344C3×C4C4×D3211yescharacteristic
345C3×C4#380211yescharacteristic
346Dic3—22no—
347Dic3—92no—
348Dic3—92no—
349Dic3—182no—
350Dic3—362no—
351Dic3—361no—
352Dic3—361no—
353C32#480311yescharacteristic
354C9—21no—
355C9#477111yescharacteristic, complemented
356C23—181no—
357C23—271no—
358Q8—32no—
359Q8—61no—
360D4—92no—
361D4—181no—
362D4—544no—
363C2×C4—21no—
364C2×C4—61no—
365C2×C4—271no—
366C2×C4—272no—
367C2×C4—272no—
368C2×C4—544no—
369C2×C4—541no—
370C2×C4—1082no—
371C2×C4C2×D3×D911yescharacteristic
372C8—61no—
373C8—181no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—362no—
378C2×C3C22⋊C4×D911yescharacteristic
379C2×C3#6347611yescharacteristic
380C2×C3#6348011yescharacteristic
381C2×C3#6392111yescharacteristic
382C2×C3#6393711yescharacteristic
383C2×C3#6395111yescharacteristic
384D3—122no—
385D3—184no—
386D3—182no—
387D3—182no—
388C22—182no—
389C22—272no—
390C22—361no—
391C22—361no—
392C22—541no—
393C22—541no—
394C22#8866811yescharacteristic
395C4—41no—
396C4—62no—
397C4—272no—
398C4—1081no—
399C4C4×D3×D911yescharacteristic
400C4#8864611yescharacteristic
401C3—21no—
402C3#1058848311yescharacteristic, complemented
403C3#1059061211yescharacteristic
404C2—362no—
405C2—542no—
406C2#1172104811yescharacteristic
407C2#1172118711yescharacteristic
408C2#1172122511yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88634C2
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C3⋊D9C2semidirect
3. C2×#88414C2semidirect
4. C9×#810C2semidirect
5. C3×#63369C2semidirect
6. #11720298C2semidirect
7. #11720283C2semidirect
17. #88634C4semidirect
38. #63369D3semidirect
50. C3×D36C2×C4semidirect
52. C9×Dic6C2×C4semidirect
118. #810D9semidirect
154. D36C4×D3semidirect
245. C3×C9#417semidirect
285. Dic6C4×D9semidirect
355. C9#4771semidirect
402. C3#10588483semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C4⋊C4×C3⋊D92
C2×#884142
C9×#8102
C3×#633692
#117202982
#117202832
#105906123
#105884833