Finite Group Databasebeta

← #13138691

Subgroups — #13138691

Order 1728.

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

normal subgroups only

This group has 516 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×#88638C211yesmaximal, characteristic
2C2×#88521C211yesmaximal, characteristic, complemented
3#11720945C211yesmaximal, characteristic, complemented
4C2×#88424C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6C3×(C2×D9)⋊C8C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590969—31nomaximal
9#10588463—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×C4×D9C2211yescharacteristic
14C2×#6194—21no—
15C2×D3×Dic9—21no—
16#88638—22no—
17#88638—22no—
18C2×C3×C8×C9C2211yescharacteristic
19#88521—22no—
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22(C2×D9)⋊Dic3—21no—
23Dic3⋊Dic9—21no—
24#88424—22no—
25C2×#3795—31no—
26C2×#3746—31no—
27#63857—31no—
28C2×#3716—31no—
29C2×#3699—31no—
30C3×#821D311yescharacteristic
31C3×(C2×D3)⋊C8—31no—
32#63567—31no—
33C2×#3672—31no—
34C2×Dic36—31no—
35C2×Dic36—61no—
36#63459—31no—
37C9×#102—31no—
38(C2×D9)⋊C8D311yescharacteristic, complemented
39#63368—31no—
40C22×D3×C9—21no—
41C22×C3×D9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9—21no—
45#6226—21no—
46#6226—21no—
47#6226—21no—
48#6226D412yes—
49C2×C9×Dic3—21no—
50C2×C3×Dic9D411yescharacteristic
51C9×D12—21no—
52C9×D12D412yes—
53C3×C4×D9—21no—
54C3×C4×D9—21no—
55#6194—24no—
56D3×Dic9—24no—
57C3×C8×C9—21no—
58C3×C9⋊C8—21no—
59#4783—91no—
60#4733—91no—
61C2×#634—31no—
62C2×#634—31no—
63C2×C3×D12C2×D311yescharacteristic
64C2×C3×C4×D3—31no—
65C2×#625—61no—
66C2×D3×Dic3—61no—
67#3795—62no—
68#3795—62no—
69C2×C32×C8C2×D311yescharacteristic
70#3746—62no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72C2×C3×C3⋊C8—31no—
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75#3716—62no—
76C2×D4×C9—31no—
77C2×#611—61no—
78C22×Dic9—61no—
79#3699—62no—
80#3699—62no—
81C2×C4×D9C2×D311yescharacteristic
82C2×Dic18—31no—
83C2×Dic18—31no—
84C2×Dic18—61no—
85C2×Dic18—61no—
86C2×C8×C9—21no—
87C2×C8×C9C2×D311yescharacteristic
88C9×C4⋊C4—31no—
89#3672—62no—
90(C2×D9)⋊C4—61no—
91C4⋊Dic9—61no—
92C2×C9⋊C8C2×D311yescharacteristic
93Dic36—62no—
94Dic36—64no—
95C22×C3×C9C2×D411yescharacteristic
96C2×D3×C9—22no—
97C2×D3×C9—41no—
98C2×D3×C9—41no—
99C2×C3×D9—21no—
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—22no—
105C3⋊Dic9—41no—
106C9×Dic3—41no—
107C3×Dic9—21no—
108C2×#237—91no—
109C2×#235—91no—
110C2×#214—91no—
111#924—91no—
112C2×Dic12—91no—
113C2×Dic12—91no—
114C2×Dic12—181no—
115C2×#204—91no—
116#897—91no—
117C3×#102—31no—
118C3×C22⋊C8—91no—
119#821D911yescharacteristic, complemented
120(C2×D3)⋊C8—31no—
121#816—91no—
122#810—91no—
123C22×C3×D3—21no—
124C22×C3×D3—31no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—61no—
128#634—32no—
129#634—61no—
130#634—61no—
131#634—61no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—31no—
134C3×D12—21no—
135C3×D12D1212yes—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138#625—64no—
139D3×Dic3—64no—
140C23×C9—61no—
141C22×D9D1211yescharacteristic
142C32×C8—21no—
143C3×C3⋊C8—61no—
144D4×C9—32no—
145D4×C9—61no—
146C2×C4×C9—21no—
147C2×C4×C9—61no—
148C2×C4×C9C22×D311yescharacteristic
149#611—64no—
150C2×Dic9—64no—
151C2×Dic9—61no—
152C2×Dic9—61no—
153C2×Dic9—121no—
154C2×Dic9—121no—
155C2×Dic9D1211yescharacteristic
156C4×D9—21no—
157C4×D9—21no—
158Dic18—32no—
159Dic18—62no—
160Dic18—61no—
161Dic18—62no—
162Dic18—61no—
163Dic18—61no—
164Dic18—121no—
165Dic18—121no—
166C8×C9—22no—
167C8×C9—21no—
168C9⋊C8—21no—
169#452—271noSylow
170C2×C3×C9#12011yescharacteristic
171C2×C3×C9#13511yescharacteristic
172C2×C3×C9#13711yescharacteristic
173D3×C9—42no—
174C3×D9—41no—
175C2×C3×D4—31no—
176C22×C3×C4—91no—
177C2×#67—181no—
178C2×#67—181no—
179C22×Dic3—181no—
180#237—182no—
181#237—182no—
182#235—182no—
183#235—182no—
184C2×D12C2×D911yescharacteristic
185C2×C4×D3—31no—
186C2×C4×D3—181no—
187C2×Dic6—91no—
188C2×Dic6—91no—
189C2×Dic6—91no—
190C2×Dic6—91no—
191C2×Dic6—181no—
192C2×Dic6—181no—
193C2×C3×C8—21no—
194C2×C3×C8—91no—
195C2×C3×C8C2×D911yescharacteristic
196C2×C3×C8D3211yescharacteristic
197C3×C4⋊C4—31no—
198#217—181no—
199#214—182no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—181no—
202C4⋊Dic3C2×D911yescharacteristic
203Dic3⋊C4—181no—
204C2×C3⋊C8—31no—
205Dic12—182no—
206Dic12—182no—
207Dic12—184no—
208#204—182no—
209C22×C32D3×D411yescharacteristic
210C2×C3×D3—22no—
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C32×C4C2×D1211yescharacteristic
217C32×C4D3×D411yescharacteristic
218C3⋊Dic3—62no—
219C3⋊Dic3—121no—
220C3×Dic3—41no—
221C3×Dic3—61no—
222C22×C9—21no—
223C22×C9—62no—
224C22×C9—121no—
225C22×C9—121no—
226C22×C9C2×D1211yescharacteristic
227C2×D9—21no—
228C2×D9—21no—
229C2×D9—21no—
230C4×C9—21no—
231C4×C9—21no—
232C4×C9—121no—
233C4×C9D3×D411yescharacteristic
234C4×C9D3×D411yescharacteristic
235Dic9—21no—
236Dic9—62no—
237Dic9—122no—
238Dic9—122no—
239Dic9—121no—
240C2×#41—271no—
241C2×Q16—271no—
242C2×SD16—271no—
243C22⋊Q8—271no—
244#103—271no—
245#102—31no—
246C22⋊C8—91no—
247C3×C9#45211yescharacteristic, complemented, Sylow
248C23×C3—61no—
249C23×C3—91no—
250C22×D3—21no—
251C22×D3—31no—
252C3×D4—32no—
253C3×D4—61no—
254C2×C3×C4—21no—
255C2×C3×C4—61no—
256C2×C3×C4—91no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4C22×D911yescharacteristic
260C2×C3×C4C2×D3211yescharacteristic
261#67—184no—
262#67—184no—
263C2×Dic3—21no—
264C2×Dic3—31no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269C2×Dic3—181no—
270C2×Dic3—361no—
271C2×Dic3—361no—
272D12—21no—
273D12D3612yes—
274C4×D3—61no—
275C4×D3—61no—
276C4×D3—184no—
277Dic6—92no—
278Dic6—92no—
279Dic6—182no—
280Dic6—181no—
281Dic6—181no—
282Dic6—182no—
283Dic6—181no—
284Dic6—181no—
285Dic6—181no—
286Dic6—181no—
287Dic6—361no—
288Dic6—361no—
289C3×C8—22no—
290C3×C8—21no—
291C3×C8—21no—
292C3×C8—181no—
293C3⋊C8—61no—
294C2×C32#88211yescharacteristic
295C2×C32#91211yescharacteristic
296C2×C32#91511yescharacteristic
297C3×D3—42no—
298C3×D3—121no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—122no—
303C2×C9#88211yescharacteristic
304C2×C9#90411yescharacteristic
305C2×C9#90911yescharacteristic
306D9—41no—
307#41—542no—
308#41—542no—
309C2×Q8—271no—
310C2×Q8—271no—
311C2×D4—31no—
312C2×D4—541no—
313C22×C4—91no—
314C22×C4—541no—
315Q16—542no—
316SD16—542no—
317C2×C8—91no—
318C2×C8D3×D911yescharacteristic
319C4⋊C4—31no—
320C4⋊C4—541no—
321C22⋊C4—541no—
322C22×C3—21no—
323C22×C3—62no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—181no—
329C22×C3D4×D911yescharacteristic
330C22×C3D3×D1211yescharacteristic
331C2×D3—22no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—61no—
335C2×D3—61no—
336C2×D3—61no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—121no—
340C3×C4—181no—
341C3×C4C2×D3611yescharacteristic
342C3×C4D4×D911yescharacteristic
343C3×C4D3×D1211yescharacteristic
344C3×C4#381111yescharacteristic
345Dic3—41no—
346Dic3—61no—
347Dic3—182no—
348Dic3—182no—
349Dic3—362no—
350Dic3—362no—
351Dic3—361no—
352Dic3—361no—
353C32#478311yescharacteristic
354C9—21no—
355C9#473311yescharacteristic, complemented
356C23—61no—
357C23—91no—
358Q8—272no—
359Q8—541no—
360Q8—541no—
361Q8—541no—
362D4—32no—
363D4—61no—
364D4—544no—
365C2×C4—61no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—544no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4C2×D3×D911yescharacteristic
373C8—21no—
374C8—181no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—122no—
379C2×C3—361no—
380C2×C3#6344411yescharacteristic
381C2×C3#6347411yescharacteristic
382C2×C3#6347711yescharacteristic
383C2×C3#6380011yescharacteristic
384C2×C3#6380711yescharacteristic
385C2×C3#6392311yescharacteristic
386D3—42no—
387D3—121no—
388C22—62no—
389C22—121no—
390C22—121no—
391C22—181no—
392C22—181no—
393C22—181no—
394C22D9×D1211yescharacteristic
395C4—121no—
396C4—181no—
397C4—542no—
398C4—1081no—
399C4D3×D3611yescharacteristic
400C4#8866811yescharacteristic
401C3—21no—
402C3#1058846311yescharacteristic, complemented
403C3#1059096911yescharacteristic
404C2—122no—
405C2—361no—
406C2#1172082211yescharacteristic
407C2#1172082711yescharacteristic
408C2#1172105111yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88638C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#452
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×#88521C2semidirect
3. #11720945C2semidirect
4. C2×#88424C2semidirect
5. C9×#821C2semidirect
6. C3×(C2×D9)⋊C8C2semidirect
7. #11720269C2semidirect
38. (C2×D9)⋊C8D3semidirect
119. #821D9semidirect
247. C3×C9#452semidirect
355. C9#4733semidirect
402. C3#10588463semidirect

Maximal subgroups

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

subgroupindex
C2×#886382
C2×#885212
#117209452
C2×#884242
C9×#8212
C3×(C2×D9)⋊C82
#117202692
#105909693
#105884633