Finite Group Databasebeta

← #13140850

Subgroups — #13140850

Order 1728.

440 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 89 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×#88548C211yesmaximal, characteristic
2C3×#63506C211yesmaximal, characteristic, complemented
3C2×C9×#216C211yesmaximal, characteristic, complemented
4#11720953C211yesmaximal, characteristic
5C2×#88427C211yesmaximal, characteristic, complemented
6C2×#88416C211yesmaximal, characteristic
7#11720274C211yesmaximal, characteristic, complemented
8#10591608—31nomaximal
9#10588870—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14#88548—22no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×#216—22no—
17C3×Dic9⋊C4—22no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20#88445—22no—
21#88427—22no—
22#88416—22no—
23C2×#3755—31no—
24C3×#944—31no—
25C2×C3×#216D311yescharacteristic
26#63864—31no—
27C2×#3718—31no—
28C2×#3717—31no—
29#63554—31no—
30C2×C9×Q16—31no—
31#63509—31no—
32#63506D311yescharacteristic, complemented
33C2×#3675—31no—
34C2×#3675—61no—
35C2×#3674—31no—
36C2×#3664—31no—
37#63375—31no—
38C22×C3⋊D9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D412yes—
48C9×Dic6—21no—
49C9×Dic6—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C3⋊(C9⋊C8)D412yes—
53C9×C3⋊C8D412yes—
54#5190—91no—
55#5086—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)—31no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6C2×D311yescharacteristic
60#3755—62no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×#216—22no—
63C3×Dic3⋊C4—62no—
64C3×C4×Dic3—31no—
65C2×C3×C3⋊C8C2×D311yescharacteristic
66#3722—62no—
67#3718—62no—
68#3717—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9—31no—
71C2×Q8×C9C2×D311yescharacteristic
72C2×D36—31no—
73C2×D36—61no—
74C2×Dic18C2×D311yescharacteristic
75C9×Q16—62no—
76C2×C8×C9—31no—
77#3675—62no—
78#3675—64no—
79#3674—62no—
80(C2×D9)⋊C4—62no—
81Dic9⋊C4—22no—
82C4×Dic9C2×D311yescharacteristic
83C2×C9⋊C8—31no—
84C2×C9⋊C8—61no—
85#3664—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—41no—
88C2×C3⋊D9—42no—
89C3×C4×C9—41no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9C2×D411yescharacteristic
92C9×Dic3—41no—
93C3×Dic9—22no—
94C3×Dic9—41no—
95C2×C3×Q16—31no—
96C3×C4⋊Q8—91no—
97#947—91no—
98#944—31no—
99C2×#216—91no—
100C2×#216—91no—
101C2×#216D911yescharacteristic, complemented
102C2×#215—91no—
103C2×#215—91no—
104C2×#215—91no—
105C2×#215—181no—
106C2×#204—91no—
107#875—91no—
108#815—91no—
109#803—91no—
110C22×C3⋊D3—61no—
111C32×Q8—21no—
112C32×Q8—21no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C4⋊(C3⋊D3)—61no—
116C4⋊(C3⋊D3)—61no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—61no—
120C3×Dic6—21no—
121C3×Dic6—21no—
122C3×Dic6—61no—
123C3×Dic6—61no—
124C22×D9—61no—
125C22×D9—121no—
126C3⋊(C3⋊C8)—32no—
127C3×C3⋊C8#6712yes—
128Q8×C9—22no—
129Q8×C9—21no—
130Q8×C9—21no—
131Q8×C9—41no—
132Q8×C9—61no—
133Q8×C9—61no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—41no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139C2×Dic9—21no—
140C2×Dic9#6712yes—
141D36—62no—
142D36—61no—
143D36—61no—
144D36—121no—
145Dic18—21no—
146Dic18—21no—
147C8×C9—32no—
148C9⋊C8—32no—
149C9⋊C8—62no—
150#497—271noSylow
151C2×C3×C9#12711yescharacteristic
152C2×C3×C9#13712yes—
153C3⋊D9—81no—
154C2×C3×Q8—21no—
155C2×C3×Q8—31no—
156C2×C3×Q8—91no—
157C2×C3×Q8C2×D911yescharacteristic
158C2×C3×Q8D3211yescharacteristic
159C2×D12—91no—
160C2×D12—91no—
161C2×D12—181no—
162C2×Dic6—31no—
163C2×Dic6C2×D911yescharacteristic
164C3×Q16—62no—
165C2×C3×C8—31no—
166C3×C4⋊C4—182no—
167C3×C42—91no—
168#216—22no—
169#216—182no—
170#216—182no—
171#215—182no—
172#215—182no—
173#215—184no—
174#215—182no—
175(C2×D3)⋊C4—182no—
176(C2×D3)⋊C4—182no—
177Dic3⋊C4—62no—
178C4×Dic3—31no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—181no—
182C2×C3⋊C8C2×D911yescharacteristic
183#204—182no—
184C22×C32D3×D411yescharacteristic
185C2×C3⋊D3—121no—
186C2×C3⋊D3—122no—
187C32×C4—41no—
188C32×C4D3×D411yescharacteristic
189C32×C4C2×#6711yescharacteristic
190C3×Dic3—41no—
191C3×Dic3—62no—
192C3×Dic3—121no—
193C22×C9—21no—
194C22×C9C2×#6711yescharacteristic
195C2×D9—122no—
196C2×D9—121no—
197C2×D9—122no—
198C2×D9—242no—
199C4×C9—21no—
200C4×C9—21no—
201C4×C9—42no—
202C4×C9—41no—
203C4×C9—121no—
204C4×C9D3×D411yescharacteristic
205C4×C9D3×D411yescharacteristic
206Dic9—22no—
207Dic9—41no—
208C2×Q16—31no—
209C2×Q16—271no—
210C2×SD16—271no—
211C2×SD16—271no—
212C4⋊Q8—91no—
213#124—271no—
214#97—271no—
215C3×C9#49711yescharacteristic, complemented, Sylow
216C22×D3—181no—
217C22×D3—181no—
218C22×D3—361no—
219C3×Q8—22no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—41no—
225C3×Q8—61no—
226C3×Q8—61no—
227C3×Q8—181no—
228C3×Q8—181no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—61no—
234C2×C3×C4—92no—
235C2×C3×C4—181no—
236C2×C3×C4C22×D911yescharacteristic
237C2×C3×C4C2×D3211yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—32no—
240C2×Dic3—61no—
241D12—182no—
242D12—181no—
243D12—181no—
244D12—181no—
245D12—181no—
246D12—361no—
247Dic6—21no—
248Dic6—21no—
249Dic6—61no—
250Dic6—61no—
251C3×C8—32no—
252C3⋊C8—92no—
253C3⋊C8—92no—
254C3⋊C8—182no—
255C3⋊C8#61112yes—
256C2×C32#91812yes—
257C2×C32#94011yescharacteristic
258C3⋊D3—241no—
259C2×C9—22no—
260C2×C9—21no—
261C2×C9#94011yescharacteristic
262C2×C9#94212yes—
263D9—242no—
264D9—241no—
265C2×Q8—31no—
266C2×Q8—91no—
267C2×Q8D3×D911yescharacteristic
268C2×D4—271no—
269Q16—62no—
270Q16—542no—
271SD16—542no—
272SD16—542no—
273C2×C8—31no—
274C2×C8—271no—
275C4⋊C4—182no—
276C22⋊C4—542no—
277C42—91no—
278C22×C3—21no—
279C22×C3D4×D911yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—362no—
282C2×D3—361no—
283C2×D3—362no—
284C2×D3—361no—
285C2×D3—362no—
286C2×D3—722no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—42no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—121no—
293C3×C4—182no—
294C3×C4—361no—
295C3×C4D4×D911yescharacteristic
296C3×C4C2×#61111yescharacteristic
297C3×C4D3×#6711yescharacteristic
298C3×C4#381111yescharacteristic
299Dic3—41no—
300Dic3—62no—
301Dic3—121no—
302C32#519011yescharacteristic
303C9—21no—
304C9#508611yescharacteristic, complemented
305C23—541no—
306Q8—21no—
307Q8—21no—
308Q8—61no—
309Q8—61no—
310Q8—181no—
311Q8—181no—
312D4—541no—
313D4—541no—
314C2×C4—21no—
315C2×C4—61no—
316C2×C4—92no—
317C2×C4—181no—
318C2×C4C2×D3×D911yescharacteristic
319C8—32no—
320C8—272no—
321C2×C3—22no—
322C2×C3—21no—
323C2×C3#6348012yes—
324C2×C3#6350211yescharacteristic
325C2×C3#6394812yes—
326C2×C3#6397911yescharacteristic
327D3—722no—
328D3—721no—
329D3—721no—
330C22—1082no—
331C22—1081no—
332C22D9×#6711yescharacteristic
333C4—41no—
334C4—121no—
335C4—182no—
336C4—361no—
337C4D3×#61111yescharacteristic
338C4#8866811yescharacteristic
339C3—21no—
340C3#1058887011yescharacteristic, complemented
341C3#1059160811yescharacteristic
342C2—2161no—
343C2#1172122212yes—
344C2#1172128911yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#497
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#63506C2semidirect
3. C2×C9×#216C2semidirect
5. C2×#88427C2semidirect
7. #11720274C2semidirect
32. #63506D3semidirect
101. C2×#216D9semidirect
215. C3×C9#497semidirect
304. C9#5086semidirect
340. C3#10588870semidirect

Maximal subgroups

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

subgroupindex
C2×#885482
C3×#635062
C2×C9×#2162
#117209532
C2×#884272
C2×#884162
#117202742
#105916083
#105888703