Finite Group Databasebeta

← #13137812

Subgroups — #13137812

Order 1728.

368 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 79 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#11720951C211yesmaximal, characteristic
3C9×#810C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic
5#11720298C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic
7#11720275C211yesmaximal, characteristic, complemented
8#10590541—31nomaximal
9#10588474—31nomaximal
10C2×#6226—21no—
11C2×C3×D36C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20(C2×D9)⋊Dic3—21no—
21#64097—31no—
22#63863—31no—
23C3×#810D311yescharacteristic
24C3×#809—31no—
25#63566—31no—
26#63563—31no—
27#63555—31no—
28#63509—31no—
29#63450—31no—
30#63450—61no—
31C9×#103—31no—
32#63395—31no—
33#63380—31no—
34#63375—31no—
35#63369D311yescharacteristic
36C22×C3×D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42#6226—42no—
43C2×C9×Dic3—21no—
44C3×D36—21no—
45C3×D36—21no—
46C9×Dic6—21no—
47C9×Dic6—21no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4794—91no—
51#4765—91no—
52C2×#634—61no—
53C2×C3×D12—31no—
54C2×C3×Dic6C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—31no—
57#3750—61no—
58C4×C3⋊Dic3—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61(C2×D3)⋊Dic3—61no—
62(C2×D3)⋊Dic3—61no—
63C2×Q8×C9—31no—
64C2×D36C2×D311yescharacteristic
65C2×Dic18—61no—
66C2×Dic18—62no—
67C2×C8×C9—31no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C2×D311yescharacteristic
70(C2×D9)⋊C4—61no—
71(C2×D9)⋊C4—61no—
72C4⋊Dic9—31no—
73C4⋊Dic9—61no—
74Dic9⋊C4—61no—
75Dic9⋊C4—62no—
76C4×Dic9—31no—
77C4×Dic9—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C2×C3×D9—41no—
81C2×C3×D9—41no—
82C2×C3×D9—41no—
83C3×C4×C9—41no—
84C3×C4×C9#4111yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—21no—
87C3⋊Dic9—21no—
88C3⋊Dic9—41no—
89C9×Dic3—41no—
90#947—91no—
91#936—91no—
92#888—91no—
93#888—91no—
94#888—181no—
95C3×#103—31no—
96C3×#102—91no—
97#835—91no—
98#832—91no—
99#821—91no—
100#816—91no—
101#815—91no—
102#815—91no—
103#810D911yescharacteristic, complemented
104#809—31no—
105C22×C3×D3—61no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—61no—
111#634—122no—
112C2×C3×Dic3—21no—
113C3×D12—61no—
114C3×D12—61no—
115C3×Dic6—21no—
116C3×Dic6—21no—
117C22×D9—21no—
118C3×C3⋊C8—21no—
119C3×C3⋊C8—61no—
120Q8×C9—61no—
121Q8×C9—61no—
122C2×C4×C9—22no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—61no—
126C2×C4×C9C22×D311yescharacteristic
127C2×Dic9—31no—
128C2×Dic9—31no—
129C2×Dic9—61no—
130C2×Dic9—61no—
131C2×Dic9—62no—
132C2×Dic9—61no—
133D36—21no—
134D36—21no—
135Dic18—124no—
136Dic18—122no—
137C8×C9—61no—
138C9⋊C8—21no—
139#489—271noSylow
140C2×C3×C9#12411yescharacteristic
141C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
142C2×C3×C9#13711yescharacteristic
143C3×D9—81no—
144C2×C3×Q8—31no—
145C2×C3×D4—91no—
146C2×D12—31no—
147C2×Dic6—181no—
148C2×Dic6—181no—
149C2×Dic6—182no—
150C2×Dic6C2×D911yescharacteristic
151C2×C3×C8—31no—
152C2×C3×C8—91no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4C2×D911yescharacteristic
155C3×C4⋊C4D3211yescharacteristic
156#217—181no—
157#217—181no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3—91no—
161C4⋊Dic3—91no—
162C4⋊Dic3—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165Dic3⋊C4—182no—
166C4×Dic3—91no—
167C4×Dic3—91no—
168C4×Dic3—181no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8C2×D911yescharacteristic
171C22×C32D3×D411yescharacteristic
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C2×C3×D3—121no—
175C32×C4—41no—
176C32×C4#23511yescharacteristic
177C32×C4#23711yescharacteristic
178C3⋊Dic3—61no—
179C3⋊Dic3—61no—
180C3⋊Dic3—121no—
181C3×Dic3—41no—
182C22×C9—21no—
183C22×C9D3×D411yescharacteristic
184C2×D9—41no—
185C2×D9—41no—
186C2×D9—41no—
187C4×C9—21no—
188C4×C9—21no—
189C4×C9—42no—
190C4×C9—41no—
191C4×C9—121no—
192C4×C9#23511yescharacteristic
193C4×C9#23711yescharacteristic
194Dic9—62no—
195Dic9—62no—
196Dic9—61no—
197Dic9—61no—
198Dic9—122no—
199Dic9—121no—
200C4⋊Q8—271no—
201#124—271no—
202#103—31no—
203#103—271no—
204#102—91no—
205#102—271no—
206#97—271no—
207C3×C9#48911yescharacteristic, complemented, Sylow
208C23×C3—181no—
209C22×D3—61no—
210C3×Q8—61no—
211C3×Q8—61no—
212C3×D4—181no—
213C3×D4—181no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—61no—
219C2×C3×C4C22×D911yescharacteristic
220C2×C3×C4C2×D3211yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—91no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—182no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231D12—61no—
232D12—61no—
233Dic6—21no—
234Dic6—21no—
235Dic6—362no—
236Dic6—364no—
237Dic6—362no—
238C3×C8—61no—
239C3×C8—181no—
240C3⋊C8—21no—
241C3⋊C8—61no—
242C2×C32#88511yescharacteristic
243C2×C32#91411yescharacteristic
244C2×C32#91811yescharacteristic
245C3×D3—241no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9—21no—
249C2×C9#88511yescharacteristic
250C2×C9#91111yescharacteristic
251C2×C9#91511yescharacteristic
252D9—81no—
253C2×Q8—31no—
254C2×Q8—541no—
255C2×D4—91no—
256C2×C8—31no—
257C2×C8—91no—
258C4⋊C4—271no—
259C4⋊C4—541no—
260C4⋊C4D3×D911yescharacteristic
261C22⋊C4—541no—
262C22⋊C4—541no—
263C42—271no—
264C22×C3—21no—
265C22×C3—361no—
266C22×C3—361no—
267C22×C3—361no—
268C22×C3D4×D911yescharacteristic
269C22×C3#381111yescharacteristic
270C2×D3—121no—
271C2×D3—121no—
272C2×D3—121no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—42no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—121no—
279C3×C4#369711yescharacteristic
280C3×C4#369911yescharacteristic
281C3×C4#379511yescharacteristic
282C3×C4#379511yescharacteristic
283Dic3—41no—
284Dic3—182no—
285Dic3—182no—
286Dic3—181no—
287Dic3—181no—
288Dic3—181no—
289Dic3—181no—
290Dic3—362no—
291Dic3—361no—
292Dic3—361no—
293C32#479411yescharacteristic
294C9—21no—
295C9#476511yescharacteristic, complemented
296C23—181no—
297Q8—61no—
298Q8—61no—
299Q8—1082no—
300D4—181no—
301D4—181no—
302C2×C4—21no—
303C2×C4—61no—
304C2×C4—271no—
305C2×C4—271no—
306C2×C4—541no—
307C2×C4C2×D3×D911yescharacteristic
308C8—61no—
309C8—181no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—721no—
314C2×C3#6344711yescharacteristic
315C2×C3#6347611yescharacteristic
316C2×C3#6348011yescharacteristic
317C2×C3#6391311yescharacteristic
318C2×C3#6393711yescharacteristic
319C2×C3#6395111yescharacteristic
320D3—241no—
321C22—361no—
322C22—361no—
323C22—361no—
324C22#8866811yescharacteristic
325C4—41no—
326C4—121no—
327C4—541no—
328C4—541no—
329C4—1081no—
330C4#8863811yescharacteristic
331C4#8864111yescharacteristic
332C3—21no—
333C3#1058847411yescharacteristic, complemented
334C3#1059054111yescharacteristic
335C2—721no—
336C2#1172104211yescharacteristic
337C2#1172118711yescharacteristic
338C2#1172122511yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#489
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×#810C2semidirect
5. #11720298C2semidirect
7. #11720275C2semidirect
103. #810D9semidirect
207. C3×C9#489semidirect
295. C9#4765semidirect
333. C3#10588474semidirect

Maximal subgroups

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

subgroupindex
#117217442
#117209512
C9×#8102
C3×#633692
#117202982
#117202832
#117202752
#105905413
#105884743