Finite Group Databasebeta

← #13138701

Subgroups — #13138701

Order 1728.

494 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 91 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×(C3⋊D9)⋊Q8C211yesmaximal, characteristic
2C2×C3×Dic36C211yesmaximal, characteristic, complemented
3#11720936C211yesmaximal, characteristic
4C2×#88427C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic
6C9×#816C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590950—31nomaximal
9#10588489—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×(C3⋊D9)⋊C4—21no—
15C2×#6189—21no—
16(C3⋊D9)⋊Q8—22no—
17(C3⋊D9)⋊Q8—22no—
18C2×C3×C8×C9C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C3×Dic36—22no—
22#88445—21no—
23Dic3⋊Dic9—21no—
24#88427—22no—
25C2×#3797—31no—
26C2×C3×Dic12—31no—
27#63871—31no—
28C2×#3718—31no—
29C2×Q8×D9—31no—
30#63647—31no—
31C3×#816D311yescharacteristic
32#63565—31no—
33C2×#3674—31no—
34C2×Dic36D311yescharacteristic, complemented
35#63459—31no—
36C9×#103—31no—
37(C2×D9)⋊C8—31no—
38(C2×D9)⋊C8—61no—
39#63368—31no—
40C22×C3⋊D9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C4×C3⋊D9—21no—
44C4×C3⋊D9—21no—
45C2×C9×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C2×C3×Dic9—21no—
49C9×Dic6—21no—
50C9×Dic6D412yes—
51C3×Dic18—21no—
52C3×Dic18—21no—
53C3×Dic18—21no—
54C3×Dic18D412yes—
55(C3⋊D9)⋊C4—24no—
56#6189—24no—
57C3×C8×C9—21no—
58C3⋊(C9⋊C8)—21no—
59#5186—91no—
60#4809—91no—
61C2×C4×C3⋊D3—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6C2×D311yescharacteristic
65C2×#627—61no—
66C2×#624—61no—
67#3797—62no—
68#3797—62no—
69C2×C32×C8C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72C3×Dic12—62no—
73Dic3⋊Dic3—61no—
74#3722—61no—
75#3718—62no—
76C2×Q8×C9—31no—
77Q8×D9—62no—
78Q8×D9—62no—
79C2×C4×D9—31no—
80C2×C4×D9—61no—
81C2×C4×D9—61no—
82C2×Dic18—61no—
83C2×Dic18C2×D311yescharacteristic
84C2×Dic18C2×D311yescharacteristic
85C2×C8×C9—21no—
86C2×C8×C9C2×D311yescharacteristic
87C9×C4⋊C4—31no—
88#3674—62no—
89(C2×D9)⋊C4—61no—
90C4⋊Dic9—61no—
91C2×C9⋊C8—31no—
92C2×C9⋊C8—61no—
93Dic36—22no—
94C22×C3×C9C2×D411yescharacteristic
95C2×C3⋊D9—21no—
96C2×C3⋊D9—21no—
97C2×C3⋊D9Q1612yescomplemented
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C3⋊Dic9—21no—
101C9×Dic3—22no—
102C9×Dic3—41no—
103C3×Dic9—22no—
104C3×Dic9—41no—
105C2×D3×Q8—91no—
106C2×D3×Q8—91no—
107C2×C3×Q16—91no—
108#946—91no—
109C2×#216—91no—
110C2×#216—91no—
111C2×Dic12—31no—
112#897—91no—
113C3×#103—31no—
114#835—91no—
115(C2×D3)⋊C8—91no—
116(C2×D3)⋊C8—91no—
117(C2×D3)⋊C8—181no—
118#816D911yescharacteristic, complemented
119#810—91no—
120C22×C3⋊D3—31no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—31no—
123C4×C3⋊D3—61no—
124C4×C3⋊D3—61no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×Dic6—21no—
130C3×Dic6—32no—
131C3×Dic6—61no—
132C3×Dic6—61no—
133C3×Dic6—61no—
134C3×Dic6D1212yes—
135#627—64no—
136#624—64no—
137C22×D9—31no—
138C22×D9—61no—
139C32×C8—21no—
140C3⋊(C3⋊C8)—61no—
141Q8×C9—32no—
142Q8×C9—61no—
143C2×C4×C9—21no—
144C2×C4×C9—61no—
145C2×C4×C9—61no—
146C2×C4×C9C22×D311yescharacteristic
147C2×Dic9—21no—
148C2×Dic9—21no—
149C2×Dic9—31no—
150C2×Dic9—61no—
151C4×D9—64no—
152C4×D9—61no—
153C4×D9—61no—
154C4×D9—121no—
155C4×D9—121no—
156Dic18—21no—
157Dic18—21no—
158Dic18—21no—
159Dic18—64no—
160Dic18#6712yes—
161C8×C9—21no—
162C8×C9—41no—
163C9⋊C8—61no—
164C9⋊C8—121no—
165#451—271noSylow
166C2×C3×C9#12011yescharacteristic
167C2×C3×C9C2×Q1611yescharacteristic
168C2×C3×C9#13711yescharacteristic
169C3⋊D9—22no—
170C2×C3×Q8—31no—
171C2×C3×Q8—91no—
172C2×C3×Q8—91no—
173D3×Q8—182no—
174D3×Q8—182no—
175D3×Q8—182no—
176D3×Q8—182no—
177C2×C4×D3—91no—
178C2×C4×D3—91no—
179C2×C4×D3—181no—
180C2×C4×D3—181no—
181C2×C4×D3—181no—
182C2×Dic6—31no—
183C2×Dic6—31no—
184C2×Dic6—181no—
185C2×Dic6—181no—
186C2×Dic6C2×D911yescharacteristic
187C3×Q16—182no—
188C2×C3×C8—21no—
189C2×C3×C8C2×D911yescharacteristic
190C2×C3×C8D3211yescharacteristic
191C3×C4⋊C4—31no—
192#216—182no—
193#216—182no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—181no—
197C4⋊Dic3C2×D911yescharacteristic
198Dic3⋊C4—181no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—181no—
202Dic12—62no—
203C22×C32D3×D411yescharacteristic
204C2×C3⋊D3—32no—
205C2×C3⋊D3—61no—
206C2×C3⋊D3—61no—
207C32×C4C2×D1211yescharacteristic
208C32×C4D3×D411yescharacteristic
209C3⋊Dic3—61no—
210C3×Dic3—22no—
211C3×Dic3—41no—
212C3×Dic3—62no—
213C3×Dic3—121no—
214C22×C9—21no—
215C22×C9D3×D411yescharacteristic
216C2×D9—32no—
217C2×D9—62no—
218C2×D9—61no—
219C2×D9—61no—
220C2×D9—121no—
221C2×D9—121no—
222C4×C9—21no—
223C4×C9—21no—
224C4×C9—62no—
225C4×C9—121no—
226C4×C9D3×D411yescharacteristic
227C4×C9C2×#6711yescharacteristic
228Dic9—22no—
229Dic9—41no—
230Dic9—61no—
231Dic9—121no—
232C22×Q8—271no—
233C2×Q16—91no—
234C2×Q16—271no—
235C22⋊Q8—271no—
236#103—31no—
237#103—271no—
238C22⋊C8—271no—
239C3×C9#45111yescharacteristic, complemented, Sylow
240C22×D3—91no—
241C22×D3—91no—
242C22×D3—181no—
243C3×Q8—32no—
244C3×Q8—61no—
245C3×Q8—92no—
246C3×Q8—181no—
247C3×Q8—181no—
248C3×Q8—181no—
249C2×C3×C4—21no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4C22×D911yescharacteristic
255C2×C3×C4C2×D3211yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—91no—
261C2×Dic3—91no—
262C2×Dic3—181no—
263C4×D3—184no—
264C4×D3—184no—
265C4×D3—181no—
266C4×D3—181no—
267C4×D3—181no—
268C4×D3—181no—
269C4×D3—361no—
270C4×D3—361no—
271Dic6—21no—
272Dic6—32no—
273Dic6—61no—
274Dic6—61no—
275Dic6—61no—
276Dic6—184no—
277Dic6—184no—
278Dic6D3612yes—
279C3×C8—21no—
280C3×C8—21no—
281C3×C8—41no—
282C3⋊C8—181no—
283C3⋊C8—181no—
284C3⋊C8—361no—
285C2×C32#88211yescharacteristic
286C2×C32#91511yescharacteristic
287C2×C32D3×Q1611yescharacteristic
288C3⋊D3—62no—
289C2×C9—21no—
290C2×C9—21no—
291C2×C9—21no—
292C2×C9D3×Q1611yescharacteristic
293C2×C9#91811yescharacteristic
294C2×C9#93711yescharacteristic
295D9—62no—
296D9—122no—
297C2×Q8—31no—
298C2×Q8—91no—
299C2×Q8—91no—
300C2×Q8—541no—
301C2×Q8—542no—
302C2×Q8—542no—
303C22×C4—271no—
304C22×C4—541no—
305Q16—182no—
306Q16—542no—
307C2×C8—271no—
308C2×C8D3×D911yescharacteristic
309C4⋊C4—31no—
310C4⋊C4—541no—
311C22⋊C4—541no—
312C22×C3—21no—
313C22×C3D4×D911yescharacteristic
314C22×C3#380211yescharacteristic
315C2×D3—92no—
316C2×D3—92no—
317C2×D3—182no—
318C2×D3—181no—
319C2×D3—181no—
320C2×D3—181no—
321C2×D3—181no—
322C2×D3—361no—
323C2×D3—361no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—62no—
327C3×C4—121no—
328C3×C4—182no—
329C3×C4—361no—
330C3×C4C2×D3611yescharacteristic
331C3×C4D4×D911yescharacteristic
332C3×C4D3×D1211yescharacteristic
333C3×C4D3×#6711yescharacteristic
334Dic3—22no—
335Dic3—41no—
336Dic3—62no—
337Dic3—121no—
338Dic3—181no—
339Dic3—181no—
340Dic3—361no—
341C32#480911yescharacteristic
342C9—21no—
343C9#518611yescharacteristic, complemented
344C23—271no—
345Q8—32no—
346Q8—61no—
347Q8—92no—
348Q8—181no—
349Q8—181no—
350Q8—181no—
351Q8—544no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—271no—
357C2×C4—544no—
358C2×C4—541no—
359C2×C4—541no—
360C2×C4C2×D3×D911yescharacteristic
361C8—21no—
362C8—541no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3#6344411yescharacteristic
367C2×C3#6347711yescharacteristic
368C2×C3Q16×D911yescharacteristic
369C2×C3#6380211yescharacteristic
370C2×C3#6381111yescharacteristic
371C2×C3#6392211yescharacteristic
372D3—182no—
373D3—182no—
374D3—362no—
375C22—272no—
376C22—541no—
377C22—541no—
378C22#8864611yescharacteristic
379C4—62no—
380C4—121no—
381C4—182no—
382C4—361no—
383C4—541no—
384C4D3×D3611yescharacteristic
385C4D9×#6711yescharacteristic
386C3—21no—
387C3#1058848911yescharacteristic, complemented
388C3#1059095011yescharacteristic
389C2—542no—
390C2#1172082511yescharacteristic
391C2#1172082911yescharacteristic
392C2#1172104911yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic36C2semidirect
4. C2×#88427C2semidirect
6. C9×#816C2semidirect
7. #11720268C2semidirect
34. C2×Dic36D3semidirect
97. C2×C3⋊D9Q16semidirect
118. #816D9semidirect
239. C3×C9#451semidirect
343. C9#5186semidirect
387. C3#10588489semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C2×C3×Dic362
#117209362
C2×#884272
#117205422
C9×#8162
#117202682
#105909503
#105884893