Finite Group Databasebeta

← #13137847

Subgroups — #13137847

Order 1728.

578 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 109 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×(C3⋊D9)⋊Q8C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic
3C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic
4C9×#810C211yesmaximal, characteristic, complemented
5C3×#63368C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10590601—31nomaximal
9D9×#103—31nomaximal
10C2×C4×C3⋊D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×(C3⋊D9)⋊C4—21no—
15C2×#6189—21no—
16(C3⋊D9)⋊Q8—22no—
17(C3⋊D9)⋊Q8C414yescomplemented
18C3×C9×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20#88534—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23(C3⋊D9)⋊C8—22no—
24C2×#3797—31no—
25C4⋊C4×C3⋊D3—31no—
26C2×#3710—31no—
27C2×Q8×D9—31no—
28C3×#810—31no—
29C3×#810D311yescharacteristic
30#63566—31no—
31#63566—31no—
32C2×C8×D9—31no—
33C4⋊C4×D9—31no—
34C4⋊C4×D9—61no—
35C9×#103—31no—
36#63395—31no—
37#63376—31no—
38#63368D311yescharacteristic, complemented
39C22×C3⋊D9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C4×C3⋊D9—42no—
45C4×C3⋊D9C2×C412yes—
46C4×C3⋊D9D412yes—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C9×Dic6—21no—
50C9×Dic6C2×C412yescomplemented
51C3×Dic18—21no—
52C3×Dic18C2×C412yescomplemented
53(C3⋊D9)⋊C4—24no—
54#6189—24no—
55C9×C3⋊C8—21no—
56C3×C9⋊C8—21no—
57D3×#103—91no—
58D3×#103—91no—
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—61no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6C2×D311yescharacteristic
63C2×#627—61no—
64C2×#624—61no—
65#3797—34no—
66#3797—62no—
67C32×C4⋊C4C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)—31no—
69#3750—61no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8C2×D311yescharacteristic
72#3710—62no—
73C2×Q8×C9—31no—
74Q8×D9—34no—
75Q8×D9—62no—
76C2×C4×D9—31no—
77C2×C4×D9—61no—
78C2×C4×D9—61no—
79C2×C4×D9—61no—
80C2×C4×D9—62no—
81C2×Dic18—61no—
82C2×Dic18C2×D311yescharacteristic
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
91C8×D9—62no—
92C22×C3×C9C2×D411yescharacteristic
93C2×C3⋊D9C22⋊C412yes—
94C2×C3⋊D9SD1612yes—
95C2×C3⋊D9Q1612yes—
96C3×C4×C9—41no—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—41no—
100C3⋊Dic9C22⋊C412yes—
101C9×Dic3—22no—
102C3×Dic9—22no—
103C2×D3×Q8—91no—
104C2×D3×Q8—91no—
105C2×D3×C8—91no—
106C2×D3×C8—91no—
107D3×C4⋊C4—91no—
108D3×C4⋊C4—91no—
109D3×C4⋊C4—181no—
110C3×#103—31no—
111C3×#103—91no—
112#835—91no—
113#835—91no—
114#816—91no—
115#816—91no—
116#810—31no—
117#810D911yescharacteristic, complemented
118C22×C3⋊D3—31no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—61no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—32no—
125C4×C3⋊D3—122no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—61no—
128C3×Dic6—21no—
129C3×Dic6—32no—
130C3×Dic6—61no—
131C3×Dic6C4×D312yes—
132#627—64no—
133#624—64no—
134C22×D9—31no—
135C22×D9—61no—
136C3×C3⋊C8—21no—
137C3×C3⋊C8—61no—
138Q8×C9—32no—
139Q8×C9—61no—
140C2×C4×C9—22no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—61no—
144C2×C4×C9C22×D311yescharacteristic
145C2×Dic9—21no—
146C2×Dic9—31no—
147C2×Dic9—61no—
148C2×Dic9—62no—
149C2×Dic9—61no—
150C4×D9—32no—
151C4×D9—32no—
152C4×D9—62no—
153C4×D9—62no—
154C4×D9—64no—
155C4×D9—122no—
156C4×D9—124no—
157Dic18—21no—
158Dic18—64no—
159Dic18C4×D312yescomplemented
160C8×C9—61no—
161C9⋊C8—21no—
162C2×#103—271noSylow
163C2×C3×C9C2×C22⋊C411yescharacteristic
164C2×C3×C9C2×SD1611yescharacteristic
165C2×C3×C9C2×Q1611yescharacteristic
166C3⋊D9#10314yescomplemented
167C2×C3×Q8—31no—
168C2×C3×Q8—91no—
169D3×Q8—94no—
170D3×Q8—94no—
171D3×Q8—182no—
172D3×Q8—182no—
173C2×C4×D3—91no—
174C2×C4×D3—91no—
175C2×C4×D3—181no—
176C2×C4×D3—181no—
177C2×C4×D3—181no—
178C2×C4×D3—181no—
179C2×C4×D3—181no—
180C2×C4×D3—182no—
181C2×Dic6—31no—
182C2×Dic6—181no—
183C2×Dic6—181no—
184C2×Dic6C2×D911yescharacteristic
185C2×C3×C8—31no—
186C2×C3×C8—91no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4C2×D911yescharacteristic
189C3×C4⋊C4D3211yescharacteristic
190C4⋊Dic3—91no—
191C4⋊Dic3—91no—
192C4⋊Dic3—181no—
193Dic3⋊C4—181no—
194Dic3⋊C4—181no—
195Dic3⋊C4—182no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8C2×D911yescharacteristic
198D3×C8—182no—
199D3×C8—182no—
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—32no—
202C2×C3⋊D3—32no—
203C2×C3⋊D3—32no—
204C32×C4—41no—
205C32×C4C2×C4×D311yescharacteristic
206C32×C4D3×D411yescharacteristic
207C3⋊Dic3—32no—
208C3⋊Dic3—121no—
209C3×Dic3—22no—
210C3×Dic3—62no—
211C22×C9—21no—
212C22×C9D3×D411yescharacteristic
213C2×D9—32no—
214C2×D9—32no—
215C2×D9—32no—
216C2×D9—62no—
217C2×D9—62no—
218C2×D9—62no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—42no—
222C4×C9—41no—
223C4×C9—62no—
224C4×C9C2×C4×D311yescharacteristic
225C4×C9D3×D411yescharacteristic
226Dic9—22no—
227Dic9—32no—
228Dic9—62no—
229Dic9—122no—
230Dic9—121no—
231C22×Q8—271no—
232C22×C8—271no—
233C2×C4⋊C4—271no—
234#103—31no—
235#103—91no—
236#103—271no—
237#103—271no—
238C3×C9C2×#10311yescharacteristic, complemented, Sylow
239C22×D3—91no—
240C22×D3—91no—
241C22×D3—181no—
242C3×Q8—32no—
243C3×Q8—61no—
244C3×Q8—92no—
245C3×Q8—181no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—61no—
251C2×C3×C4—181no—
252C2×C3×C4C22×D911yescharacteristic
253C2×C3×C4C2×D3211yescharacteristic
254C2×Dic3—21no—
255C2×Dic3—61no—
256C2×Dic3—91no—
257C2×Dic3—91no—
258C2×Dic3—181no—
259C2×Dic3—182no—
260C2×Dic3—181no—
261C2×Dic3—181no—
262C4×D3—92no—
263C4×D3—92no—
264C4×D3—92no—
265C4×D3—92no—
266C4×D3—182no—
267C4×D3—182no—
268C4×D3—184no—
269C4×D3—184no—
270C4×D3—362no—
271C4×D3—362no—
272C4×D3—364no—
273Dic6—21no—
274Dic6—32no—
275Dic6—61no—
276Dic6—184no—
277Dic6—184no—
278Dic6C4×D912yescomplemented
279C3×C8—61no—
280C3×C8—181no—
281C3⋊C8—21no—
282C3⋊C8—61no—
283C2×C32D3×C22⋊C411yescharacteristic
284C2×C32D3×SD1611yescharacteristic
285C2×C32D3×Q1611yescharacteristic
286C3⋊D3—34no—
287C2×C9—21no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9D3×C22⋊C411yescharacteristic
291C2×C9D3×SD1611yescharacteristic
292C2×C9D3×Q1611yescharacteristic
293D9—34no—
294D9—64no—
295C2×Q8—31no—
296C2×Q8—91no—
297C2×Q8—274no—
298C2×Q8—541no—
299C2×Q8—542no—
300C22×C4—271no—
301C22×C4—541no—
302C22×C4—541no—
303C2×C8—31no—
304C2×C8—91no—
305C2×C8—542no—
306C4⋊C4—271no—
307C4⋊C4—541no—
308C4⋊C4D3×D911yescharacteristic
309C22×C3—21no—
310C22×C3D4×D911yescharacteristic
311C22×C3#381111yescharacteristic
312C2×D3—92no—
313C2×D3—92no—
314C2×D3—92no—
315C2×D3—92no—
316C2×D3—92no—
317C2×D3—92no—
318C2×D3—182no—
319C2×D3—182no—
320C2×D3—182no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—42no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—62no—
327C3×C4—182no—
328C3×C4C2×C4×D911yescharacteristic
329C3×C4D4×D911yescharacteristic
330C3×C4C4×D3211yescharacteristic
331C3×C4#380211yescharacteristic
332Dic3—22no—
333Dic3—62no—
334Dic3—92no—
335Dic3—92no—
336Dic3—182no—
337Dic3—362no—
338Dic3—361no—
339Dic3—361no—
340C32D3×#10311yescharacteristic
341C9—21no—
342C9D3×#10311yescharacteristic, complemented
343C23—271no—
344Q8—32no—
345Q8—61no—
346Q8—92no—
347Q8—181no—
348Q8—544no—
349C2×C4—21no—
350C2×C4—61no—
351C2×C4—181no—
352C2×C4—271no—
353C2×C4—272no—
354C2×C4—272no—
355C2×C4—544no—
356C2×C4—541no—
357C2×C4—1082no—
358C2×C4C2×D3×D911yescharacteristic
359C8—61no—
360C8—181no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3C22⋊C4×D911yescharacteristic
365C2×C3SD16×D911yescharacteristic
366C2×C3Q16×D911yescharacteristic
367C2×C3#6392111yescharacteristic
368C2×C3#6393011yescharacteristic
369C2×C3#6394411yescharacteristic
370D3—94no—
371D3—94no—
372D3—184no—
373C22—272no—
374C22—272no—
375C22—272no—
376C22#8866811yescharacteristic
377C4—41no—
378C4—62no—
379C4—182no—
380C4—272no—
381C4—1081no—
382C4C4×D3×D911yescharacteristic
383C4#8864611yescharacteristic
384C3—21no—
385C3D9×#10311yescharacteristic, complemented
386C3#1059060111yescharacteristic
387C2—274no—
388C2#1172104811yescharacteristic
389C2#1172118211yescharacteristic
390C2#1172122011yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C2×#103
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
4. C9×#810C2semidirect
5. C3×#63368C2semidirect
6. #11720298C2semidirect
7. #11720276C2semidirect
17. (C3⋊D9)⋊Q8C4semidirect
38. #63368D3semidirect
50. C9×Dic6C2×C4semidirect
52. C3×Dic18C2×C4semidirect
117. #810D9semidirect
159. Dic18C4×D3semidirect
166. C3⋊D9#103semidirect
238. C3×C9C2×#103semidirect
278. Dic6C4×D9semidirect
342. C9D3×#103semidirect
385. C3D9×#103semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C4⋊C4×C3⋊D92
C2×(C3⋊D9)⋊C82
C9×#8102
C3×#633682
#117202982
#117202762
#105906013
D9×#1033