Finite Group Databasebeta

← #13143979

Subgroups — #13143979

Order 1728.

790 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 157 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×Q8×C3⋊D9C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3C2×#88524C211yesmaximal, characteristic, complemented
4#11721751C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6#11720542C211yesmaximal, characteristic, complemented
7#11720527C211yesmaximal, characteristic, complemented
8#10592505—31nomaximal
9#10588486—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11Q8×C3⋊D9—22no—
12Q8×C3⋊D9—22no—
13C2×C4⋊(C3⋊D9)—21no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9—21no—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×#6226—21no—
18C2×#6226C2211yescharacteristic
19C2×C3×C8×C9C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21#88548—22no—
22#88536—21no—
23C2×C3⋊(C9⋊C8)C2211yescharacteristic
24#88524—22no—
25C2×Q8×C3⋊D3—31no—
26C2×#3755—31no—
27C2×#3744—31no—
28#64104—31no—
29C2×Q8×D9—33no—
30C32×#103D311yescharacteristic
31#63647—31no—
32#63637—31no—
33C2×#3675—33no—
34C2×#3664—33no—
35#63457—33no—
36C9×#103D313yescomplemented
37(C2×D9)⋊C8—33no—
38#63368—33no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9D412yescomplemented
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9D411yescharacteristic
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)—21no—
50C4⋊(C3⋊D9)—42no—
51C4×C3⋊D9—24no—
52C4×C3⋊D9—21no—
53C4×C3⋊D9—21no—
54#6226—24no—
55#6226—21no—
56#6226D412yescomplemented
57C3×C8×C9—21no—
58C3⋊(C9⋊C8)—21no—
59#4806—93no—
60#4806—91no—
61C2×C32×Q8C2×D311yescharacteristic
62Q8×C3⋊D3—62no—
63Q8×C3⋊D3—62no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—61no—
66C2×C4×C3⋊D3—31no—
67C2×C4×C3⋊D3—61no—
68C2×#634—31no—
69C2×#634—61no—
70C2×C32×C8C2×D311yescharacteristic
71C32×C4⋊C4C2×D311yescharacteristic
72#3755—62no—
73#3752—61no—
74C2×C3⋊(C3⋊C8)—31no—
75#3744—62no—
76C2×Q8×C9C2×D313yes—
77Q8×D9—66no—
78Q8×D9—66no—
79C2×D36—33no—
80C2×D36—63no—
81C2×C4×D9—33no—
82C2×C4×D9—63no—
83C2×Dic18—33no—
84C2×Dic18—63no—
85C2×C8×C9C2×D313yes—
86C9×C4⋊C4C2×D313yes—
87#3675—66no—
88(C2×D9)⋊C4—63no—
89C2×C9⋊C8—33no—
90#3664—66no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9—21no—
93C2×C3⋊D9—21no—
94C2×C3⋊D9—41no—
95C2×C3⋊D9—41no—
96C2×C3⋊D9—41no—
97C2×C3⋊D9SD1612yescomplemented
98C3×C4×C9—22no—
99C3×C4×C9—41no—
100C3×C4×C9C2×D411yescharacteristic
101C3×C4×C9C2×D411yescharacteristic
102C3⋊Dic9—21no—
103C3⋊Dic9—22no—
104C2×D3×Q8—93no—
105C2×D3×Q8—91no—
106C2×#215—93no—
107C2×#215—91no—
108C2×#204—93no—
109C2×#204—91no—
110#895—93no—
111#895—91no—
112C3×#103D913yescomplemented
113C3×#103C3⋊D311yescharacteristic
114(C2×D3)⋊C8—93no—
115(C2×D3)⋊C8—91no—
116#810—93no—
117#810—91no—
118C22×C3⋊D3—31no—
119C22×C3⋊D3—61no—
120C32×Q8—21no—
121C32×Q8D1212yes—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—61no—
127C4⋊(C3⋊D3)—61no—
128C4⋊(C3⋊D3)—61no—
129C4⋊(C3⋊D3)—122no—
130C4×C3⋊D3—64no—
131C4×C3⋊D3—61no—
132C4×C3⋊D3—61no—
133#634—32no—
134#634—64no—
135#634—61no—
136C22×D9—33no—
137C22×D9—63no—
138C32×C8—21no—
139C3⋊(C3⋊C8)—61no—
140Q8×C9—23no—
141Q8×C9D1216yescomplemented
142C2×C4×C9—23no—
143C2×C4×C9—23no—
144C2×C4×C9C22×D313yes—
145C2×Dic9—33no—
146C2×Dic9—63no—
147D36—63no—
148D36—63no—
149D36—126no—
150C4×D9—612no—
151C4×D9—63no—
152C4×D9—63no—
153Dic18—36no—
154Dic18—612no—
155Dic18—63no—
156C8×C9—23no—
157C9⋊C8—63no—
158#448—271noSylow
159C2×C3×C9#12011yescharacteristic
160C2×C3×C9C2×SD1611yescharacteristic
161C2×C3×C9#13711yescharacteristic
162C3⋊D9—22no—
163C3⋊D9—81no—
164C2×C3×Q8C2×D913yes—
165C2×C3×Q8C2×C3⋊D311yescharacteristic
166D3×Q8—182no—
167D3×Q8—186no—
168D3×Q8—186no—
169D3×Q8—182no—
170C2×D12—93no—
171C2×D12—91no—
172C2×D12—181no—
173C2×D12—183no—
174C2×C4×D3—93no—
175C2×C4×D3—91no—
176C2×C4×D3—181no—
177C2×C4×D3—183no—
178C2×Dic6—93no—
179C2×Dic6—91no—
180C2×Dic6—181no—
181C2×Dic6—183no—
182C2×C3×C8C2×D913yes—
183C2×C3×C8C2×C3⋊D311yescharacteristic
184C3×C4⋊C4C2×D913yes—
185C3×C4⋊C4C2×C3⋊D311yescharacteristic
186#215—186no—
187#215—182no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—183no—
190C2×C3⋊C8—93no—
191C2×C3⋊C8—91no—
192#204—182no—
193#204—186no—
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—32no—
196C2×C3⋊D3—61no—
197C2×C3⋊D3—61no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C32×C4—22no—
202C32×C4—41no—
203C32×C4C2×D1211yescharacteristic
204C32×C4D3×D411yescharacteristic
205C3⋊Dic3—62no—
206C3⋊Dic3—61no—
207C22×C9D3×D413yes—
208C2×D9—36no—
209C2×D9—63no—
210C2×D9—63no—
211C2×D9—123no—
212C2×D9—123no—
213C2×D9—123no—
214C4×C9—26no—
215C4×C9—43no—
216C4×C9C2×D1213yes—
217C4×C9D3×D413yes—
218Dic9—66no—
219Dic9—63no—
220C22×Q8—271no—
221C2×SD16—271no—
222C2×SD16—271no—
223#121—271no—
224#103—271no—
225#103C3⋊D911yescharacteristic, complemented
226C22⋊C8—271no—
227C3×C9#44811yescharacteristic, complemented, Sylow
228C22×D3—93no—
229C22×D3—91no—
230C22×D3—183no—
231C22×D3—181no—
232C3×Q8—23no—
233C3×Q8—21no—
234C3×Q8D3616yescomplemented
235C3×Q8C4⋊(C3⋊D3)12yes—
236C2×C3×C4—23no—
237C2×C3×C4—21no—
238C2×C3×C4—23no—
239C2×C3×C4—21no—
240C2×C3×C4C22×D913yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242C2×Dic3—93no—
243C2×Dic3—91no—
244C2×Dic3—183no—
245C2×Dic3—181no—
246D12—183no—
247D12—183no—
248D12—181no—
249D12—181no—
250D12—362no—
251D12—366no—
252C4×D3—184no—
253C4×D3—1812no—
254C4×D3—183no—
255C4×D3—183no—
256C4×D3—181no—
257C4×D3—181no—
258Dic6—96no—
259Dic6—92no—
260Dic6—184no—
261Dic6—1812no—
262Dic6—183no—
263Dic6—181no—
264C3×C8—23no—
265C3×C8—21no—
266C3⋊C8—183no—
267C3⋊C8—181no—
268C2×C32#88211yescharacteristic
269C2×C32D3×SD1611yescharacteristic
270C2×C32#91811yescharacteristic
271C3⋊D3—62no—
272C3⋊D3—241no—
273C2×C9#88213yes—
274C2×C9D3×SD1613yes—
275C2×C9#91813yes—
276D9—66no—
277D9—243no—
278C2×Q8—271no—
279C2×Q8—541no—
280C2×Q8—542no—
281C2×Q8—542no—
282C2×Q8C2×C3⋊D911yescharacteristic
283C2×D4—271no—
284C2×D4—541no—
285C22×C4—271no—
286C22×C4—541no—
287SD16—542no—
288SD16—542no—
289C2×C8—271no—
290C2×C8C2×C3⋊D911yescharacteristic
291C4⋊C4C2×C3⋊D911yescharacteristic
292C22⋊C4—541no—
293C22×C3D4×D913yes—
294C22×C3D4×C3⋊D311yescharacteristic
295C2×D3—92no—
296C2×D3—96no—
297C2×D3—181no—
298C2×D3—181no—
299C2×D3—183no—
300C2×D3—183no—
301C2×D3—361no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—363no—
305C2×D3—363no—
306C2×D3—363no—
307C3×C4—22no—
308C3×C4—26no—
309C3×C4—41no—
310C3×C4—43no—
311C3×C4C2×D3613yes—
312C3×C4D4×D913yes—
313C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
314C3×C4D4×C3⋊D311yescharacteristic
315Dic3—182no—
316Dic3—181no—
317Dic3—186no—
318Dic3—183no—
319C32#480611yescharacteristic
320C9#480613yescomplemented
321C23—271no—
322C23—541no—
323Q8—21no—
324Q8—272no—
325Q8—544no—
326Q8—541no—
327Q8C4⋊(C3⋊D9)12yescomplemented
328D4—541no—
329D4—541no—
330D4—1082no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—271no—
334C2×C4—544no—
335C2×C4—541no—
336C2×C4—541no—
337C2×C4—541no—
338C2×C4C22×C3⋊D911yescharacteristic
339C8—21no—
340C8—541no—
341C2×C3#6344413yes—
342C2×C3SD16×D913yes—
343C2×C3#6348013yes—
344C2×C3#6409111yescharacteristic
345C2×C3SD16×C3⋊D311yescharacteristic
346C2×C3#6412711yescharacteristic
347D3—186no—
348D3—182no—
349D3—723no—
350D3—721no—
351C22—272no—
352C22—541no—
353C22—541no—
354C22—1081no—
355C22—1081no—
356C22—1081no—
357C22D4×C3⋊D911yescharacteristic
358C4—22no—
359C4—41no—
360C4—541no—
361C4—542no—
362C4C2×C4⋊(C3⋊D9)11yescharacteristic
363C4D4×C3⋊D911yescharacteristic
364C3#1058848613yescomplemented
365C3#1059250511yescharacteristic
366C2—542no—
367C2—2161no—
368C2#1172172811yescharacteristic
369C2SD16×C3⋊D911yescharacteristic
370C2#1172180411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radical#11721751C2
hypercenter#103C3⋊D9
hypocenterC3×C9#448
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊D9C2semidirect
2. C2×#88548C2semidirect
3. C2×#88524C2semidirect
5. C3×C9×#103C2semidirect
6. #11720542C2semidirect
7. #11720527C2semidirect
36. C9×#103D3semidirect
42. C3×Q8×C9D4semidirect
56. #6226D4semidirect
97. C2×C3⋊D9SD16semidirect
112. C3×#103D9semidirect
141. Q8×C9D12semidirect
225. #103C3⋊D9semidirect
227. C3×C9#448semidirect
234. C3×Q8D36semidirect
320. C9#4806semidirect
327. Q8C4⋊(C3⋊D9)semidirect
364. C3#10588486semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D92
C2×#885482
C2×#885242
#117217512
C3×C9×#1032
#117205422
#117205272
#105925053
#105884863