Finite Group Databasebeta

← #13144750

Subgroups — #13144750

Order 1728.

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

normal subgroups only

This group has 740 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
2C2×C3×C9×Q16C211yesmaximal, characteristic, complemented
3#11721856C211yesmaximal, characteristic
4C2×#88547C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720542C211yesmaximal, characteristic
7#11720539C211yesmaximal, characteristic, complemented
8#10592885—31nomaximal
9#10588866—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×Q8×C9C2211yescharacteristic
12Q8×C3⋊D9—22no—
13Q8×C3⋊D9—22no—
14C2×C4×C3⋊D9—21no—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×#6226—21no—
17C2×#6226C2211yescharacteristic
18C3×C9×Q16—22no—
19C2×C3×C8×C9C2211yescharacteristic
20#88547—22no—
21#88536—21no—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534—21no—
24C2×C3⋊(C9⋊C8)C2211yescharacteristic
25C2×Q8×C3⋊D3—31no—
26C2×C32×Q16D311yescharacteristic
27#64155—31no—
28C2×#3756—31no—
29C2×Q8×D9—33no—
30#63662—31no—
31#63647—31no—
32#63643—31no—
33C2×C9×Q16D313yescomplemented
34#63508—33no—
35C2×#3674—33no—
36#63395—33no—
37(C2×D9)⋊C8—33no—
38#63376—33no—
39C22×C3⋊D9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C3×Q8×C9—21no—
43C3×Q8×C9D412yes—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9D411yescharacteristic
50C4×C3⋊D9—24no—
51C4×C3⋊D9—21no—
52C4×C3⋊D9—21no—
53#6226—24no—
54#6226—21no—
55#6226D412yes—
56C3×C8×C9—21no—
57C3⋊(C9⋊C8)—21no—
58#5186—93no—
59#5186—91no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C2×D311yescharacteristic
62Q8×C3⋊D3—62no—
63Q8×C3⋊D3—62no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—61no—
66C2×#634—31no—
67C2×#634—61no—
68C32×Q16—22no—
69C2×C32×C8C2×D311yescharacteristic
70#3756—62no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—31no—
73#3750—61no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×Q8×C9C2×D313yes—
76C2×Q8×C9C2×D313yes—
77Q8×D9—66no—
78Q8×D9—66no—
79C2×C4×D9—33no—
80C2×C4×D9—63no—
81C2×Dic18—33no—
82C2×Dic18—63no—
83C9×Q16—26no—
84C2×C8×C9C2×D313yes—
85#3674—66no—
86(C2×D9)⋊C4—63no—
87C4⋊Dic9—33no—
88Dic9⋊C4—63no—
89C2×C9⋊C8—33no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—21no—
92C2×C3⋊D9—21no—
93C2×C3⋊D9Q1612yescomplemented
94C3×C4×C9—22no—
95C3×C4×C9—41no—
96C3×C4×C9C2×D411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—21no—
99C3⋊Dic9—22no—
100C3⋊Dic9—41no—
101C2×D3×Q8—93no—
102C2×D3×Q8—91no—
103C2×C3×Q16D913yescomplemented
104C2×C3×Q16C3⋊D311yescharacteristic
105#946—93no—
106#946—91no—
107C2×#216—93no—
108C2×#216—91no—
109#835—93no—
110#835—91no—
111(C2×D3)⋊C8—93no—
112(C2×D3)⋊C8—91no—
113#816—93no—
114#816—91no—
115C22×C3⋊D3—31no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8#6712yes—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—61no—
126C4×C3⋊D3—64no—
127C4×C3⋊D3—61no—
128C4×C3⋊D3—61no—
129#634—32no—
130#634—64no—
131#634—61no—
132C22×D9—33no—
133C32×C8—21no—
134C3⋊(C3⋊C8)—61no—
135Q8×C9—23no—
136Q8×C9—23no—
137Q8×C9—23no—
138Q8×C9#6716yes—
139C2×C4×C9—23no—
140C2×C4×C9—23no—
141C2×C4×C9C22×D313yes—
142C2×Dic9—33no—
143C2×Dic9—63no—
144C2×Dic9—63no—
145C4×D9—612no—
146C4×D9—63no—
147C4×D9—63no—
148Dic18—36no—
149Dic18—612no—
150Dic18—63no—
151C8×C9—23no—
152C9⋊C8—63no—
153#451—271noSylow
154C2×C3×C9#12011yescharacteristic
155C2×C3×C9C2×Q1611yescharacteristic
156C2×C3×C9#13711yescharacteristic
157C3⋊D9—22no—
158C2×C3×Q8C2×D913yes—
159C2×C3×Q8C2×D913yes—
160C2×C3×Q8C2×C3⋊D311yescharacteristic
161C2×C3×Q8C2×C3⋊D311yescharacteristic
162D3×Q8—182no—
163D3×Q8—186no—
164D3×Q8—186no—
165D3×Q8—182no—
166C2×C4×D3—93no—
167C2×C4×D3—91no—
168C2×C4×D3—181no—
169C2×C4×D3—183no—
170C2×Dic6—93no—
171C2×Dic6—91no—
172C2×Dic6—181no—
173C2×Dic6—183no—
174C3×Q16—26no—
175C3×Q16—22no—
176C2×C3×C8C2×D913yes—
177C2×C3×C8C2×C3⋊D311yescharacteristic
178#216—182no—
179#216—186no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—183no—
182C4⋊Dic3—93no—
183C4⋊Dic3—91no—
184Dic3⋊C4—181no—
185Dic3⋊C4—183no—
186C2×C3⋊C8—93no—
187C2×C3⋊C8—91no—
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—32no—
190C2×C3⋊D3—61no—
191C2×C3⋊D3—61no—
192C32×C4—22no—
193C32×C4—41no—
194C32×C4D3×D411yescharacteristic
195C32×C4C2×#6711yescharacteristic
196C3⋊Dic3—62no—
197C3⋊Dic3—61no—
198C3⋊Dic3—121no—
199C22×C9D3×D413yes—
200C2×D9—36no—
201C2×D9—63no—
202C2×D9—63no—
203C4×C9—26no—
204C4×C9—43no—
205C4×C9D3×D413yes—
206C4×C9C2×#6713yes—
207Dic9—66no—
208Dic9—63no—
209Dic9—123no—
210C22×Q8—271no—
211C2×Q16—271no—
212C2×Q16C3⋊D911yescharacteristic, complemented
213C22⋊Q8—271no—
214#103—271no—
215#103—271no—
216C22⋊C8—271no—
217C3×C9#45111yescharacteristic, complemented, Sylow
218C22×D3—93no—
219C22×D3—91no—
220C3×Q8—23no—
221C3×Q8—21no—
222C3×Q8—23no—
223C3×Q8—23no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8#61116yes—
227C3×Q8#63812yes—
228C2×C3×C4—23no—
229C2×C3×C4—21no—
230C2×C3×C4—23no—
231C2×C3×C4—21no—
232C2×C3×C4C22×D913yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×Dic3—93no—
235C2×Dic3—91no—
236C2×Dic3—183no—
237C2×Dic3—181no—
238C2×Dic3—183no—
239C2×Dic3—181no—
240C4×D3—184no—
241C4×D3—1812no—
242C4×D3—183no—
243C4×D3—183no—
244C4×D3—181no—
245C4×D3—181no—
246Dic6—96no—
247Dic6—92no—
248Dic6—184no—
249Dic6—1812no—
250Dic6—183no—
251Dic6—181no—
252C3×C8—23no—
253C3×C8—21no—
254C3⋊C8—183no—
255C3⋊C8—181no—
256C2×C32D3×Q1611yescharacteristic
257C2×C32#91811yescharacteristic
258C2×C32#93711yescharacteristic
259C3⋊D3—62no—
260C2×C9D3×Q1613yes—
261C2×C9#91813yes—
262C2×C9#93713yes—
263D9—66no—
264C2×Q8—271no—
265C2×Q8—541no—
266C2×Q8—542no—
267C2×Q8—542no—
268C2×Q8C2×C3⋊D911yescharacteristic
269C2×Q8C2×C3⋊D911yescharacteristic
270C22×C4—271no—
271C22×C4—541no—
272Q16—22no—
273Q16—542no—
274C2×C8—271no—
275C2×C8C2×C3⋊D911yescharacteristic
276C4⋊C4—271no—
277C4⋊C4—541no—
278C22⋊C4—541no—
279C22×C3D4×D913yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—92no—
282C2×D3—96no—
283C2×D3—183no—
284C2×D3—181no—
285C2×D3—181no—
286C2×D3—183no—
287C3×C4—26no—
288C3×C4—22no—
289C3×C4—43no—
290C3×C4—41no—
291C3×C4D4×D913yes—
292C3×C4C2×#61113yes—
293C3×C4D4×C3⋊D311yescharacteristic
294C3×C4C2×#63811yescharacteristic
295Dic3—186no—
296Dic3—183no—
297Dic3—182no—
298Dic3—181no—
299Dic3—361no—
300Dic3—363no—
301C32#518611yescharacteristic
302C9#518613yescomplemented
303C23—271no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—272no—
308Q8—544no—
309Q8—541no—
310Q8#623312yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—271no—
314C2×C4—544no—
315C2×C4—541no—
316C2×C4—541no—
317C2×C4—541no—
318C2×C4—541no—
319C2×C4C22×C3⋊D911yescharacteristic
320C8—21no—
321C8—541no—
322C2×C3Q16×D913yes—
323C2×C3#6348013yes—
324C2×C3#6349913yes—
325C2×C3Q16×C3⋊D311yescharacteristic
326C2×C3#6412711yescharacteristic
327C2×C3#6414611yescharacteristic
328D3—182no—
329D3—186no—
330C22—272no—
331C22—541no—
332C22—541no—
333C22D4×C3⋊D911yescharacteristic
334C4—22no—
335C4—41no—
336C4—541no—
337C4—542no—
338C4—1081no—
339C4D4×C3⋊D911yescharacteristic
340C4C2×#623311yescharacteristic
341C3#1058886613yescomplemented
342C3#1059288511yescharacteristic
343C2—542no—
344C2Q16×C3⋊D911yescharacteristic
345C2#1172180411yescharacteristic
346C2#1172183711yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C3×C9×Q16C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC2×Q16C3⋊D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×Q16C2semidirect
4. C2×#88547C2semidirect
5. #11720582C2semidirect
7. #11720539C2semidirect
33. C2×C9×Q16D3semidirect
93. C2×C3⋊D9Q16semidirect
103. C2×C3×Q16D9semidirect
212. C2×Q16C3⋊D9semidirect
217. C3×C9#451semidirect
302. C9#5186semidirect
341. C3#10588866semidirect

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×C3×C9×Q162
#117218562
C2×#885472
#117205822
#117205422
#117205392
#105928853
#105888663