Finite Group Databasebeta

← #13137758

Subgroups — #13137758

Order 1728.

422 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#11721744C211yesmaximal, characteristic, complemented
2#11721043C211yesmaximal, characteristic
3C9×#809C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720271C211yesmaximal, characteristic, complemented
8#10590552—31nomaximal
9#10588440—31nomaximal
10C2×#6226—21no—
11C2×C9×D12C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×#6194—21no—
14C2×#6194—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17#88534—21no—
18C4×C3⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#64097—31no—
22#63914—31no—
23C3×#809—31no—
24C3×#809D311yescharacteristic
25#63563—31no—
26#63563—31no—
27#63553—31no—
28#63502—31no—
29#63450—31no—
30#63450—61no—
31C9×#102—31no—
32#63392—31no—
33#63380—31no—
34C8×Dic9—31no—
35#63369D311yescharacteristic, complemented
36C22×D3×C9—21no—
37C22×C3×D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43#6226—42no—
44C9×D12—21no—
45C9×D12—21no—
46C3×D36—21no—
47C3×D36—21no—
48#6194—42no—
49#6194—42no—
50C9×C3⋊C8—21no—
51C3×C9⋊C8—21no—
52#4760—91no—
53#4760—91no—
54C2×#634—61no—
55C2×C3×D12—31no—
56C2×C3×D12C2×D311yescharacteristic
57C2×#625—61no—
58C2×#625—61no—
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)—31no—
61#3750—61no—
62C4×C3⋊Dic3—31no—
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65C2×D4×C9—31no—
66C2×#611—61no—
67C2×#611—61no—
68C2×D36C2×D311yescharacteristic
69C2×Dic18—61no—
70C2×Dic18—62no—
71C2×C8×C9—31no—
72C9×C4⋊C4—21no—
73C9×C4⋊C4C2×D311yescharacteristic
74C4⋊Dic9—31no—
75C4⋊Dic9—61no—
76Dic9⋊C4—61no—
77Dic9⋊C4—62no—
78C4×Dic9—31no—
79C4×Dic9—61no—
80C2×C9⋊C8C2×D311yescharacteristic
81C22×C3×C9C2×D411yescharacteristic
82C2×D3×C9—41no—
83C2×D3×C9—41no—
84C2×D3×C9—41no—
85C2×C3×D9—41no—
86C2×C3×D9—41no—
87C2×C3×D9—41no—
88C3×C4×C9—41no—
89C3×C4×C9#4111yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C3⋊Dic9—41no—
92C3⋊Dic9D812yes—
93C3⋊Dic9SD1612yes—
94#940—91no—
95#940—91no—
96#888—91no—
97#888—91no—
98#888—181no—
99C3×#102—31no—
100C3×#102—91no—
101#832—91no—
102#832—91no—
103#821—91no—
104#821—91no—
105C8×Dic3—91no—
106C8×Dic3—91no—
107#809—31no—
108#809D911yescharacteristic, complemented
109C22×C3×D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—61no—
116#634—122no—
117C3×D12—21no—
118C3×D12—21no—
119C3×D12—61no—
120C3×D12—61no—
121#625—122no—
122#625—122no—
123C23×C9—61no—
124C22×D9—21no—
125C3×C3⋊C8—21no—
126C3×C3⋊C8—61no—
127D4×C9—61no—
128D4×C9—61no—
129C2×C4×C9—22no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9C22×D311yescharacteristic
133#611—122no—
134#611—122no—
135C2×Dic9—31no—
136C2×Dic9—31no—
137C2×Dic9—61no—
138C2×Dic9—61no—
139C2×Dic9—62no—
140C2×Dic9—61no—
141D36—21no—
142D36—21no—
143Dic18—122no—
144Dic18—124no—
145C8×C9—61no—
146C9⋊C8—21no—
147#486—271noSylow
148C2×C3×C9#12411yescharacteristic
149C2×C3×C9C2×D811yescharacteristic
150C2×C3×C9C2×SD1611yescharacteristic
151D3×C9—81no—
152C3×D9—81no—
153C2×C3×D4—31no—
154C2×C3×D4—91no—
155C2×#67—181no—
156C2×#67—181no—
157C2×#67—181no—
158C2×#67—181no—
159C2×D12—31no—
160C2×D12C2×D911yescharacteristic
161C2×Dic6—181no—
162C2×Dic6—181no—
163C2×Dic6—182no—
164C2×C3×C8—31no—
165C2×C3×C8—91no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4C2×D911yescharacteristic
168C3×C4⋊C4D3211yescharacteristic
169C4⋊Dic3—91no—
170C4⋊Dic3—91no—
171C4⋊Dic3—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—182no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C4×Dic3—181no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8C2×D911yescharacteristic
180C22×C32D3×D411yescharacteristic
181C2×C3×D3—41no—
182C2×C3×D3—41no—
183C2×C3×D3—41no—
184C2×C3×D3—121no—
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C32×C4—41no—
188C32×C4#23511yescharacteristic
189C32×C4#23711yescharacteristic
190C3⋊Dic3—32no—
191C3⋊Dic3—32no—
192C3⋊Dic3—121no—
193C22×C9—21no—
194C22×C9—121no—
195C22×C9—121no—
196C22×C9—121no—
197C22×C9D3×D411yescharacteristic
198C2×D9—41no—
199C2×D9—41no—
200C2×D9—41no—
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—42no—
204C4×C9—41no—
205C4×C9#23511yescharacteristic
206C4×C9#23711yescharacteristic
207Dic9—32no—
208Dic9—32no—
209Dic9—62no—
210Dic9—62no—
211Dic9—122no—
212Dic9—121no—
213C4⋊Q8—271no—
214#127—271no—
215#102—31no—
216#102—91no—
217#102—271no—
218#102—271no—
219C4×C8—271no—
220C3×C9#48611yescharacteristic, complemented, Sylow
221C23×C3—61no—
222C23×C3—181no—
223C22×D3—21no—
224C22×D3—61no—
225C3×D4—61no—
226C3×D4—61no—
227C3×D4—181no—
228C3×D4—181no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4C22×D911yescharacteristic
234C2×C3×C4C2×D3211yescharacteristic
235#67—362no—
236#67—362no—
237#67—362no—
238#67—362no—
239C2×Dic3—91no—
240C2×Dic3—91no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245C2×Dic3—182no—
246C2×Dic3—181no—
247C2×Dic3—181no—
248D12—21no—
249D12—21no—
250D12—61no—
251D12—61no—
252Dic6—362no—
253Dic6—362no—
254Dic6—364no—
255C3×C8—61no—
256C3×C8—181no—
257C3⋊C8—21no—
258C3⋊C8—61no—
259C2×C32#88511yescharacteristic
260C2×C32D3×D811yescharacteristic
261C2×C32D3×SD1611yescharacteristic
262C3×D3—81no—
263C3×D3—241no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—21no—
267C2×C9—241no—
268C2×C9#88511yescharacteristic
269C2×C9D3×D811yescharacteristic
270C2×C9D3×SD1611yescharacteristic
271D9—81no—
272C2×Q8—541no—
273C2×D4—31no—
274C2×D4—91no—
275C2×D4—541no—
276C2×D4—541no—
277C2×C8—31no—
278C2×C8—91no—
279C4⋊C4—271no—
280C4⋊C4—541no—
281C4⋊C4D3×D911yescharacteristic
282C42—271no—
283C22×C3—21no—
284C22×C3—121no—
285C22×C3—121no—
286C22×C3—121no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3—361no—
290C22×C3D4×D911yescharacteristic
291C22×C3#381111yescharacteristic
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—121no—
296C2×D3—121no—
297C2×D3—121no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—42no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4#369711yescharacteristic
304C3×C4#369911yescharacteristic
305C3×C4#379511yescharacteristic
306C3×C4#379511yescharacteristic
307Dic3—92no—
308Dic3—92no—
309Dic3—92no—
310Dic3—92no—
311Dic3—182no—
312Dic3—182no—
313Dic3—362no—
314Dic3—361no—
315Dic3—361no—
316C32#476011yescharacteristic
317C9—21no—
318C9#476011yescharacteristic, complemented
319C23—61no—
320C23—181no—
321Q8—1082no—
322D4—61no—
323D4—61no—
324D4—181no—
325D4—181no—
326D4—1082no—
327D4—1082no—
328C2×C4—21no—
329C2×C4—271no—
330C2×C4—271no—
331C2×C4—541no—
332C2×C4C2×D3×D911yescharacteristic
333C8—61no—
334C8—181no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—241no—
339C2×C3—721no—
340C2×C3#6344711yescharacteristic
341C2×C3D8×D911yescharacteristic
342C2×C3SD16×D911yescharacteristic
343C2×C3#6391311yescharacteristic
344C2×C3#6392611yescharacteristic
345C2×C3#6394011yescharacteristic
346D3—81no—
347D3—241no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22#8866811yescharacteristic
355C4—41no—
356C4—272no—
357C4—272no—
358C4—1081no—
359C4#8863811yescharacteristic
360C4#8864111yescharacteristic
361C3—21no—
362C3#1058844011yescharacteristic, complemented
363C3#1059055211yescharacteristic
364C2—241no—
365C2—721no—
366C2#1172104211yescharacteristic
367C2#1172117411yescharacteristic
368C2#1172121211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721043C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#486
LCSG ▷ C2×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
1. #11721744C2semidirect
3. C9×#809C2semidirect
4. C3×#63369C2semidirect
5. #11720295C2semidirect
6. #11720283C2semidirect
7. #11720271C2semidirect
35. #63369D3semidirect
108. #809D9semidirect
220. C3×C9#486semidirect
318. C9#4760semidirect
362. C3#10588440semidirect

Maximal subgroups

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

subgroupindex
#117217442
#117210432
C9×#8092
C3×#633692
#117202952
#117202832
#117202712
#105905523
#105884403