Finite Group Databasebeta

← #13138714

Subgroups — #13138714

Order 1728.

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

normal subgroups only

This group has 524 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×#88635C211yesmaximal, characteristic
2C2×#88524C211yesmaximal, characteristic, complemented
3#11720945C211yesmaximal, characteristic, complemented
4C2×#88428C211yesmaximal, characteristic, complemented
5C3×(C2×D9)⋊C8C211yesmaximal, characteristic, complemented
6C9×#816C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590970—31nomaximal
9#10588488—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×#6192—21no—
15C2×(C3⋊D9)⋊C4—21no—
16#88635—22no—
17#88635—22no—
18C2×C3×C8×C9C2211yescharacteristic
19#88524—22no—
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22(C2×D9)⋊Dic3—21no—
23Dic3⋊Dic9—21no—
24#88428—22no—
25C2×#3799—31no—
26C2×#3744—31no—
27#63857—31no—
28C2×#3719—31no—
29C2×#3701—31no—
30C3×(C2×D3)⋊C8—31no—
31C3×#816D311yescharacteristic
32#63564—31no—
33C2×#3674—31no—
34C2×#3664—31no—
35C2×#3664—61no—
36#63459—31no—
37C9×#103—31no—
38(C2×D9)⋊C8D311yescharacteristic, complemented
39#63369—31no—
40C22×C3⋊D9—21no—
41C22×C3×D9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)D412yes—
46#6226—21no—
47#6226—21no—
48C2×C9×Dic3—21no—
49C2×C9×Dic3—21no—
50C2×C3×Dic9D411yescharacteristic
51C3×C4×D9—21no—
52C3×C4×D9—21no—
53C9×Dic6—21no—
54C9×Dic6D412yes—
55#6192—24no—
56(C3⋊D9)⋊C4—24no—
57C3×C8×C9—21no—
58C3×C9⋊C8—21no—
59#4808—91no—
60#4733—91no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×#634—31no—
63C2×C3×C4×D3—31no—
64C2×C3×Dic6C2×D311yescharacteristic
65C2×#626—61no—
66C2×#624—61no—
67#3799—62no—
68#3799—62no—
69C2×C32×C8C2×D311yescharacteristic
70#3744—62no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72C2×C3×C3⋊C8—31no—
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75#3719—62no—
76C2×Q8×C9—31no—
77#3701—62no—
78#3701—62no—
79C2×D36—31no—
80C2×D36—61no—
81C2×D36—61no—
82C2×C4×D9—61no—
83C2×C4×D9C2×D311yescharacteristic
84C2×Dic18—31no—
85C2×Dic18—61no—
86C2×C8×C9—21no—
87C2×C8×C9C2×D311yescharacteristic
88C9×C4⋊C4—31no—
89#3674—62no—
90(C2×D9)⋊C4—61no—
91C4⋊Dic9—61no—
92C2×C9⋊C8C2×D311yescharacteristic
93#3664—62no—
94#3664—64no—
95C22×C3×C9C2×D411yescharacteristic
96C2×C3⋊D9—22no—
97C2×C3⋊D9—41no—
98C2×C3⋊D9—41no—
99C2×C3×D9—21no—
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—41no—
105C9×Dic3—22no—
106C9×Dic3—41no—
107C3×Dic9—21no—
108C2×#239—91no—
109C2×#235—91no—
110C2×#216—91no—
111#924—91no—
112C2×Dic12—91no—
113C2×#204—91no—
114C2×#204—91no—
115C2×#204—181no—
116#897—91no—
117C3×#103—31no—
118C3×C22⋊C8—91no—
119#821—91no—
120(C2×D3)⋊C8—31no—
121#816D911yescharacteristic, complemented
122#809—91no—
123C22×C3⋊D3—61no—
124C22×C3×D3—31no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—61no—
127C4⋊(C3⋊D3)—32no—
128C4⋊(C3⋊D3)—61no—
129#634—61no—
130#634—61no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—31no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×Dic6—21no—
137C3×Dic6D1212yes—
138#626—64no—
139#624—64no—
140C22×D9—61no—
141C22×D9—121no—
142C22×D9D1211yescharacteristic
143C32×C8—21no—
144C3×C3⋊C8—61no—
145Q8×C9—32no—
146Q8×C9—61no—
147C2×C4×C9—21no—
148C2×C4×C9—61no—
149C2×C4×C9—61no—
150C2×C4×C9C22×D311yescharacteristic
151C2×Dic9—61no—
152C2×Dic9—121no—
153C2×Dic9D1211yescharacteristic
154D36—32no—
155D36—62no—
156D36—64no—
157D36—61no—
158D36—121no—
159C4×D9—21no—
160C4×D9—21no—
161C4×D9—64no—
162Dic18—62no—
163Dic18—61no—
164Dic18—61no—
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
173C3⋊D9—42no—
174C3×D9—41no—
175C2×C3×Q8—31no—
176C22×C3×C4—91no—
177C2×#67—181no—
178#239—182no—
179#239—182no—
180#235—182no—
181#235—182no—
182C2×D12—91no—
183C2×D12—91no—
184C2×D12—181no—
185C2×D12—181no—
186C2×C4×D3—31no—
187C2×C4×D3—181no—
188C2×C4×D3—181no—
189C2×Dic6—91no—
190C2×Dic6—91no—
191C2×Dic6—181no—
192C2×Dic6C2×D911yescharacteristic
193C2×C3×C8—21no—
194C2×C3×C8—91no—
195C2×C3×C8C2×D911yescharacteristic
196C2×C3×C8D3211yescharacteristic
197C3×C4⋊C4—31no—
198#217—181no—
199#216—182no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—181no—
202C4⋊Dic3C2×D911yescharacteristic
203Dic3⋊C4—181no—
204C2×C3⋊C8—31no—
205Dic12—182no—
206#204—182no—
207#204—182no—
208#204—184no—
209C22×C32D3×D411yescharacteristic
210C2×C3⋊D3—62no—
211C2×C3⋊D3—121no—
212C2×C3⋊D3—121no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C32×C4C2×D1211yescharacteristic
217C32×C4D3×D411yescharacteristic
218C3⋊Dic3—121no—
219C3×Dic3—22no—
220C3×Dic3—41no—
221C3×Dic3—61no—
222C22×C9—21no—
223C22×C9C2×D1211yescharacteristic
224C2×D9—21no—
225C2×D9—21no—
226C2×D9—21no—
227C2×D9—62no—
228C2×D9—122no—
229C2×D9—121no—
230C2×D9—121no—
231C2×D9—241no—
232C2×D9—241no—
233C4×C9—21no—
234C4×C9—21no—
235C4×C9—62no—
236C4×C9—121no—
237C4×C9D3×D411yescharacteristic
238C4×C9D3×D411yescharacteristic
239Dic9—21no—
240Dic9—122no—
241Dic9—121no—
242C2×#41—271no—
243C2×Q16—271no—
244C2×SD16—271no—
245C22⋊Q8—271no—
246#103—31no—
247#102—271no—
248C22⋊C8—91no—
249C3×C9#45211yescharacteristic, complemented, Sylow
250C23×C3—91no—
251C22×D3—31no—
252C22×D3—181no—
253C22×D3—181no—
254C22×D3—361no—
255C3×Q8—32no—
256C3×Q8—61no—
257C2×C3×C4—21no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—91no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4C22×D911yescharacteristic
264C2×C3×C4C2×D3211yescharacteristic
265#67—184no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—31no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C2×Dic3—361no—
272D12—92no—
273D12—92no—
274D12—182no—
275D12—184no—
276D12—181no—
277D12—181no—
278D12—361no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—184no—
282C4×D3—184no—
283Dic6—21no—
284Dic6—182no—
285Dic6—181no—
286Dic6—181no—
287Dic6—181no—
288Dic6—181no—
289Dic6—361no—
290Dic6D3612yes—
291C3×C8—22no—
292C3×C8—21no—
293C3×C8—21no—
294C3×C8—181no—
295C3⋊C8—61no—
296C2×C32#88211yescharacteristic
297C2×C32#91611yescharacteristic
298C2×C32#91811yescharacteristic
299C3⋊D3—122no—
300C3×D3—121no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9#88211yescharacteristic
305C2×C9#90411yescharacteristic
306C2×C9#90911yescharacteristic
307D9—41no—
308D9—122no—
309D9—242no—
310#41—542no—
311#41—542no—
312C2×Q8—31no—
313C2×Q8—271no—
314C2×D4—271no—
315C2×D4—541no—
316C22×C4—91no—
317C22×C4—541no—
318Q16—542no—
319SD16—542no—
320C2×C8—91no—
321C2×C8D3×D911yescharacteristic
322C4⋊C4—31no—
323C4⋊C4—541no—
324C22⋊C4—541no—
325C22×C3—21no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—181no—
329C22×C3D4×D911yescharacteristic
330C22×C3D3×D1211yescharacteristic
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—61no—
334C2×D3—182no—
335C2×D3—182no—
336C2×D3—362no—
337C2×D3—361no—
338C2×D3—361no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—721no—
342C2×D3—721no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—62no—
346C3×C4—121no—
347C3×C4—181no—
348C3×C4C2×D3611yescharacteristic
349C3×C4D4×D911yescharacteristic
350C3×C4D3×D1211yescharacteristic
351C3×C4#381111yescharacteristic
352Dic3—22no—
353Dic3—41no—
354Dic3—61no—
355Dic3—362no—
356Dic3—361no—
357Dic3—361no—
358C32#480811yescharacteristic
359C9—21no—
360C9#473311yescharacteristic, complemented
361C23—91no—
362C23—541no—
363Q8—32no—
364Q8—61no—
365Q8—541no—
366Q8—541no—
367D4—272no—
368D4—544no—
369D4—541no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—544no—
376C2×C4—541no—
377C2×C4C2×D3×D911yescharacteristic
378C8—21no—
379C8—181no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—361no—
384C2×C3#6344411yescharacteristic
385C2×C3#6347811yescharacteristic
386C2×C3#6348011yescharacteristic
387C2×C3#6380111yescharacteristic
388C2×C3#6380611yescharacteristic
389C2×C3#6392311yescharacteristic
390D3—121no—
391D3—362no—
392D3—362no—
393D3—722no—
394C22—181no—
395C22—181no—
396C22—181no—
397C22—542no—
398C22—1081no—
399C22—1081no—
400C22D9×D1211yescharacteristic
401C4—62no—
402C4—121no—
403C4—181no—
404C4—1081no—
405C4D3×D3611yescharacteristic
406C4#8866811yescharacteristic
407C3—21no—
408C3#1058848811yescharacteristic, complemented
409C3#1059097011yescharacteristic
410C2—361no—
411C2—1082no—
412C2#1172082111yescharacteristic
413C2#1172082811yescharacteristic
414C2#1172105111yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88635C2
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×#88524C2semidirect
3. #11720945C2semidirect
4. C2×#88428C2semidirect
5. C3×(C2×D9)⋊C8C2semidirect
6. C9×#816C2semidirect
7. #11720267C2semidirect
38. (C2×D9)⋊C8D3semidirect
121. #816D9semidirect
249. C3×C9#452semidirect
360. C9#4733semidirect
408. C3#10588488semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
C2×#885242
#117209452
C2×#884282
C3×(C2×D9)⋊C82
C9×#8162
#117202672
#105909703
#105884883