Finite Group Databasebeta

← #13138475

Subgroups — #13138475

Order 1728.

542 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 95 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×D3×Dic18C211yesmaximal, characteristic, complemented
2C2×#88524C211yesmaximal, characteristic, complemented
3#11721038C211yesmaximal, characteristic
4C2×#88416C211yesmaximal, characteristic, complemented
5C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C3×#63376C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10590951—31nomaximal
9#10588404—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C4×D3×C9C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×#6195—21no—
15C2×D3×Dic9—21no—
16C2×#6189—21no—
17D3×Dic18—22no—
18D3×Dic18—22no—
19C2×C3×C8×C9C2211yescharacteristic
20#88524—22no—
21C3×C4⋊Dic9C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23#88445—21no—
24#88416—22no—
25C2×D3×Dic6—31no—
26C2×#3744—31no—
27#63909—31no—
28C2×#3717—31no—
29C22×Dic18—31no—
30C3×(C2×D3)⋊C8D311yescharacteristic
31C3×#816—31no—
32#63567—31no—
33#63492—31no—
34C2×#3664—31no—
35C2×#3664—31no—
36C2×#3664—61no—
37C9×C22⋊C8—31no—
38#63376—31no—
39#63376D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×D3×C9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)—21no—
46#6226—21no—
47#6226D412yescomplemented
48C2×C9×Dic3D411yescharacteristic
49C2×C3×Dic9—21no—
50C2×C3×Dic9—21no—
51C4×D3×C9—21no—
52C4×D3×C9—21no—
53C3×Dic18—21no—
54C3×Dic18D412yescomplemented
55#6195—42no—
56D3×Dic9—24no—
57#6189—24no—
58C3×C8×C9—21no—
59C9×C3⋊C8—21no—
60#4806—91no—
61#4738—91no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×#634—31no—
64C2×C3×C4×D3C2×D311yescharacteristic
65C2×C3×Dic6—31no—
66C2×#627—61no—
67C2×#626—61no—
68C2×D3×Dic3—61no—
69D3×Dic6—62no—
70D3×Dic6—62no—
71C2×C32×C8C2×D311yescharacteristic
72#3744—62no—
73C3×C4⋊Dic3—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3722—61no—
76#3717—62no—
77C22×C4×C9—31no—
78C2×#611—61no—
79C22×Dic9—61no—
80C2×D36—31no—
81C2×D36—61no—
82C2×Dic18—31no—
83C2×Dic18—62no—
84C2×Dic18—61no—
85C2×Dic18—62no—
86C2×Dic18—61no—
87C2×Dic18C2×D311yescharacteristic
88C2×C8×C9—21no—
89C2×C8×C9—31no—
90C2×C8×C9C2×D311yescharacteristic
91(C2×D9)⋊C4—61no—
92C4⋊Dic9C2×D311yescharacteristic
93#3664—62no—
94#3664—64no—
95#3664—62no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—41no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×D3×C9—21no—
101C2×D3×C9—21no—
102C2×D3×C9SD1612yescomplemented
103C3×C4×C9C2×D411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C3⋊Dic9—22no—
106C9×Dic3—21no—
107C3×Dic9—22no—
108C3×Dic9—41no—
109C2×D3×Q8—91no—
110C22×Dic6—91no—
111C2×#215—91no—
112#930—91no—
113C2×#204—91no—
114C2×#204—91no—
115C2×#204—91no—
116C2×#204—181no—
117#895—91no—
118C3×#103—91no—
119C3×C22⋊C8—31no—
120(C2×D3)⋊C8D911yescharacteristic, complemented
121#816—31no—
122#816—91no—
123#810—91no—
124C22×C3⋊D3—61no—
125C22×C3×D3D1211yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—61no—
128C4⋊(C3⋊D3)—61no—
129C4⋊(C3⋊D3)—61no—
130#634—32no—
131#634—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3D1211yescharacteristic
135C3×C4×D3—21no—
136C3×C4×D3—21no—
137C3×Dic6—32no—
138C3×Dic6—61no—
139#627—64no—
140#626—122no—
141D3×Dic3—64no—
142C23×C9—31no—
143C22×D9—61no—
144C22×D9—121no—
145C32×C8—21no—
146C3×C3⋊C8—21no—
147C2×C4×C9—21no—
148C2×C4×C9—31no—
149C2×C4×C9—61no—
150C2×C4×C9—61no—
151C2×C4×C9C22×D311yescharacteristic
152#611—122no—
153C2×Dic9—21no—
154C2×Dic9—21no—
155C2×Dic9—64no—
156C2×Dic9—61no—
157C2×Dic9—121no—
158D36—62no—
159D36—61no—
160D36—61no—
161D36—121no—
162Dic18—21no—
163Dic18—32no—
164Dic18—62no—
165Dic18—64no—
166Dic18—61no—
167Dic18—121no—
168Dic18D1212yescomplemented
169C8×C9—22no—
170C8×C9—21no—
171C8×C9—61no—
172#448—271noSylow
173C2×C3×C9#12011yescharacteristic
174C2×C3×C9C2×SD1611yescharacteristic
175C2×C3×C9#13711yescharacteristic
176C3⋊D9—81no—
177D3×C9—22no—
178C2×C3×Q8—91no—
179C22×C3×C4—31no—
180C2×#67—181no—
181C22×Dic3—181no—
182D3×Q8—182no—
183D3×Q8—182no—
184C2×D12—91no—
185C2×D12—91no—
186C2×D12—181no—
187C2×D12—181no—
188C2×C4×D3—181no—
189C2×C4×D3C2×D911yescharacteristic
190C2×Dic6—31no—
191C2×Dic6—91no—
192C2×Dic6—91no—
193C2×Dic6—181no—
194C2×Dic6—182no—
195C2×Dic6—182no—
196C2×Dic6—181no—
197C2×Dic6—181no—
198C2×C3×C8—21no—
199C2×C3×C8—31no—
200C2×C3×C8C2×D911yescharacteristic
201C2×C3×C8D3211yescharacteristic
202C3×C4⋊C4—91no—
203#215—182no—
204(C2×D3)⋊C4—181no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—31no—
207C2×C3⋊C8C2×D911yescharacteristic
208#204—182no—
209#204—182no—
210#204—184no—
211#204—182no—
212C22×C32C2×D1211yescharacteristic
213C2×C3⋊D3—121no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—121no—
216C2×C3×D3—21no—
217C2×C3×D3—21no—
218C2×C3×D3#20412yes—
219C32×C4D3×D411yescharacteristic
220C32×C4D3×D411yescharacteristic
221C3⋊Dic3—62no—
222C3×Dic3—21no—
223C3×Dic3—62no—
224C3×Dic3—121no—
225C22×C9—21no—
226C22×C9—32no—
227C22×C9—61no—
228C22×C9—61no—
229C22×C9D3×D411yescharacteristic
230C2×D9—122no—
231C2×D9—121no—
232C2×D9—121no—
233C2×D9—121no—
234C2×D9—241no—
235C2×D9—241no—
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—61no—
239C4×C9C2×D1211yescharacteristic
240C4×C9D3×D411yescharacteristic
241Dic9—22no—
242Dic9—41no—
243Dic9—62no—
244Dic9—122no—
245C22×Q8—271no—
246C2×SD16—271no—
247C2×SD16—271no—
248#121—271no—
249#103—91no—
250#103—271no—
251C22⋊C8—31no—
252C3×C9#44811yescharacteristic, complemented, Sylow
253C23×C3—31no—
254C22×D3—181no—
255C22×D3—181no—
256C22×D3—361no—
257C22×D3D3611yescharacteristic
258C3×Q8—92no—
259C3×Q8—181no—
260C2×C3×C4—21no—
261C2×C3×C4—31no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4C22×D911yescharacteristic
267C2×C3×C4C2×D3211yescharacteristic
268#67—362no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—184no—
272C2×Dic3—181no—
273C2×Dic3—181no—
274C2×Dic3—361no—
275C2×Dic3D3611yescharacteristic
276D12—182no—
277D12—181no—
278D12—181no—
279D12—181no—
280D12—181no—
281D12—361no—
282D12—362no—
283C4×D3—21no—
284C4×D3—21no—
285C4×D3—184no—
286Dic6—32no—
287Dic6—61no—
288Dic6—92no—
289Dic6—92no—
290Dic6—182no—
291Dic6—184no—
292Dic6—184no—
293Dic6—181no—
294Dic6—181no—
295Dic6—361no—
296C3×C8—22no—
297C3×C8—21no—
298C3×C8—21no—
299C3×C8—61no—
300C3⋊C8—21no—
301C2×C32#88211yescharacteristic
302C2×C32C2×#20411yescharacteristic
303C2×C32#90911yescharacteristic
304C3⋊D3—241no—
305C3×D3—22no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—62no—
310C2×C9#88211yescharacteristic
311C2×C9D3×SD1611yescharacteristic
312C2×C9#91811yescharacteristic
313D9—242no—
314D9—241no—
315C2×Q8—91no—
316C2×Q8—271no—
317C2×Q8—541no—
318C2×Q8—542no—
319C2×Q8—542no—
320C2×D4—271no—
321C2×D4—541no—
322C22×C4—31no—
323C22×C4—541no—
324SD16—542no—
325SD16—542no—
326C2×C8—31no—
327C2×C8D3×D911yescharacteristic
328C4⋊C4—91no—
329C22⋊C4—541no—
330C22×C3—21no—
331C22×C3—32no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3C2×D3611yescharacteristic
335C22×C3D3×D1211yescharacteristic
336C2×D3—21no—
337C2×D3—21no—
338C2×D3—362no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—361no—
345C2×D3—721no—
346C2×D3—721no—
347C2×D3#366412yescomplemented
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—61no—
351C3×C4—182no—
352C3×C4—361no—
353C3×C4D4×D911yescharacteristic
354C3×C4D4×D911yescharacteristic
355C3×C4D3×D1211yescharacteristic
356C3×C4#381111yescharacteristic
357Dic3—21no—
358Dic3—62no—
359Dic3—121no—
360Dic3—182no—
361Dic3—182no—
362Dic3—362no—
363C32#473811yescharacteristic
364C9—21no—
365C9#480611yescharacteristic, complemented
366C23—31no—
367C23—541no—
368Q8—92no—
369Q8—181no—
370Q8—272no—
371Q8—544no—
372Q8—541no—
373D4—541no—
374D4—541no—
375D4—1082no—
376C2×C4—31no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—544no—
382C2×C4—541no—
383C2×C4C2×D3×D911yescharacteristic
384C8—21no—
385C8—61no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—62no—
390C2×C3#6344411yescharacteristic
391C2×C3C2×#366411yescharacteristic
392C2×C3#6347111yescharacteristic
393C2×C3D3×#20411yescharacteristic
394C2×C3#6380111yescharacteristic
395C2×C3#6392311yescharacteristic
396D3—22no—
397D3—722no—
398D3—721no—
399D3—721no—
400C22—32no—
401C22—61no—
402C22—61no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22D3×D3611yescharacteristic
407C4—61no—
408C4—182no—
409C4—361no—
410C4—542no—
411C4D9×D1211yescharacteristic
412C4#8866811yescharacteristic
413C3—21no—
414C3#1058840411yescharacteristic, complemented
415C3#1059095111yescharacteristic
416C2—62no—
417C2—2161no—
418C2#1172079711yescharacteristic
419C2D3×#366411yescharacteristic
420C2#1172105111yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radical#11721038C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#448
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
1. C2×D3×Dic18C2semidirect
2. C2×#88524C2semidirect
4. C2×#88416C2semidirect
5. C9×(C2×D3)⋊C8C2semidirect
6. C3×#63376C2semidirect
7. #11720277C2semidirect
39. #63376D3semidirect
47. #6226D4semidirect
54. C3×Dic18D4semidirect
102. C2×D3×C9SD16semidirect
120. (C2×D3)⋊C8D9semidirect
168. Dic18D12semidirect
252. C3×C9#448semidirect
347. C2×D3#3664semidirect
365. C9#4806semidirect
414. C3#10588404semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C2×#885242
#117210382
C2×#884162
C9×(C2×D3)⋊C82
C3×#633762
#117202772
#105909513
#105884043