Finite Group Databasebeta

← #13138219

Subgroups — #13138219

Order 1728.

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

normal subgroups only

This group has 496 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×#88525C211yesmaximal, characteristic, complemented
2#11721006C211yesmaximal, characteristic, complemented
3#11720993C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5C3×#63380C211yesmaximal, characteristic, complemented
6C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11720284C211yesmaximal, characteristic, complemented
8#10590765—31nomaximal
9#10588527—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×#6192—21no—
14C2×#6192—21no—
15C2×(C3⋊D9)⋊C4—21no—
16C2×C3×C8×C9C2211yescharacteristic
17#88525—22no—
18C4×C9×Dic3C2211yescharacteristic
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21#88445—21no—
22#88417—22no—
23C2×#3745—31no—
24#63911—31no—
25#63894—31no—
26C2×#3714—31no—
27C3×#821—31no—
28C3×Dic3⋊C8D311yescharacteristic
29#63564—31no—
30C2×D72—31no—
31C2×D72—31no—
32C2×D72—61no—
33#63436—31no—
34C4×D36—31no—
35C9×C4⋊C8—31no—
36#63380—31no—
37#63380D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3×D9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)D412yescomplemented
46C2×C9×Dic3D411yescharacteristic
47C2×C9×Dic3D411yescharacteristic
48C2×C3×Dic9—21no—
49C3×D36—21no—
50C3×D36—21no—
51#6192—42no—
52#6192—42no—
53(C3⋊D9)⋊C4—42no—
54C3×C8×C9—21no—
55C9×C3⋊C8—21no—
56#4843—91no—
57#4786—91no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C3×D12—31no—
61C2×#626—61no—
62C2×#626—61no—
63C2×#624—61no—
64C2×C32×C8C2×D311yescharacteristic
65#3745—62no—
66C3×C4⋊Dic3—31no—
67C3×C4×Dic3C2×D311yescharacteristic
68C2×C3×C3⋊C8C2×D311yescharacteristic
69#3722—61no—
70#3714—62no—
71C2×D36—31no—
72C2×D36—31no—
73C2×D36—61no—
74C2×D36—61no—
75C2×D36—61no—
76C2×D36—61no—
77C2×D36C2×D311yescharacteristic
78C2×C4×D9—61no—
79C2×C8×C9—21no—
80C2×C8×C9—31no—
81C2×C8×C9C2×D311yescharacteristic
82C42×C9—31no—
83(C2×D9)⋊C4—61no—
84C4⋊Dic9C2×D311yescharacteristic
85D72—62no—
86D72—64no—
87D72—62no—
88C22×C3×C9C2×D411yescharacteristic
89C2×C3⋊D9—22no—
90C2×C3⋊D9—41no—
91C2×C3⋊D9—41no—
92C2×C3⋊D9—41no—
93C2×C3⋊D9—41no—
94C2×C3⋊D9—41no—
95C2×C3×D9—41no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9#4111yescharacteristic
100C9×Dic3—21no—
101C9×Dic3D812yescomplemented
102C3×Dic9—41no—
103#940—91no—
104C2×#213—91no—
105C2×D24—91no—
106C2×D24—91no—
107C2×D24—91no—
108C2×D24—181no—
109#893—91no—
110#874—91no—
111C4×D12—91no—
112C3×C4⋊C8—31no—
113C3×#102—91no—
114#821—31no—
115#821—91no—
116Dic3⋊C8D911yescharacteristic, complemented
117#809—91no—
118C22×C3⋊D3—61no—
119C22×C3⋊D3—61no—
120C22×C3×D3—61no—
121C2×C32×C4C22×D311yescharacteristic
122C4⋊(C3⋊D3)—32no—
123C4⋊(C3⋊D3)—61no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3D1211yescharacteristic
128C2×C3×Dic3D1211yescharacteristic
129C3×D12—61no—
130C3×D12—61no—
131#626—122no—
132#626—122no—
133#624—122no—
134C22×D9—21no—
135C22×D9—61no—
136C22×D9—61no—
137C22×D9—121no—
138C22×D9—121no—
139C32×C8—21no—
140C3×C3⋊C8—21no—
141C2×C4×C9—21no—
142C2×C4×C9—31no—
143C2×C4×C9—31no—
144C2×C4×C9C22×D311yescharacteristic
145C2×Dic9—21no—
146D36—21no—
147D36—21no—
148D36—32no—
149D36—62no—
150D36—62no—
151D36—61no—
152D36—61no—
153D36—61no—
154D36—121no—
155D36—121no—
156D36—122no—
157D36—122no—
158C4×D9—122no—
159C8×C9—22no—
160C8×C9—21no—
161C8×C9—61no—
162#459—271noSylow
163C2×C3×C9#12111yescharacteristic
164C2×C3×C9C2×D811yescharacteristic
165C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
166C3⋊D9—42no—
167C3⋊D9—81no—
168C3×D9—81no—
169C2×C3×D4—91no—
170C2×#67—181no—
171C2×#67—181no—
172C2×D12—31no—
173C2×D12—91no—
174C2×D12—91no—
175C2×D12—91no—
176C2×D12—91no—
177C2×D12—181no—
178C2×D12—181no—
179C2×D12—181no—
180C2×D12—181no—
181C2×C4×D3—181no—
182C2×C4×D3—181no—
183C2×C3×C8—21no—
184C2×C3×C8—31no—
185C2×C3×C8C2×D911yescharacteristic
186C2×C3×C8D3211yescharacteristic
187C3×C4⋊C4—91no—
188C3×C42—31no—
189#213—182no—
190(C2×D3)⋊C4—181no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—31no—
193C4×Dic3C2×D911yescharacteristic
194C2×C3⋊C8C2×D911yescharacteristic
195D24—182no—
196D24—182no—
197D24—184no—
198D24—182no—
199C22×C32C2×D1211yescharacteristic
200C2×C3⋊D3—62no—
201C2×C3⋊D3—121no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C32×C4D3×D411yescharacteristic
210C32×C4#23911yescharacteristic
211C3×Dic3—21no—
212C3×Dic3—121no—
213C3×Dic3D2412yes—
214C22×C9—21no—
215C22×C9D3×D411yescharacteristic
216C2×D9—41no—
217C2×D9—41no—
218C2×D9—41no—
219C2×D9—62no—
220C2×D9—122no—
221C2×D9—121no—
222C2×D9—121no—
223C2×D9—122no—
224C2×D9—121no—
225C2×D9—121no—
226C2×D9—121no—
227C2×D9—241no—
228C2×D9—241no—
229C2×D9—241no—
230C2×D9—241no—
231C4×C9—21no—
232C4×C9—21no—
233C4×C9—32no—
234C4×C9—61no—
235C4×C9#23511yescharacteristic
236C4×C9D3×D411yescharacteristic
237Dic9—41no—
238C2×D8—271no—
239C2×D8—271no—
240#127—271no—
241C4×D4—271no—
242C4⋊C8—31no—
243#102—91no—
244#102—271no—
245C3×C9#45911yescharacteristic, complemented, Sylow
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—181no—
249C22×D3—181no—
250C22×D3—181no—
251C22×D3—181no—
252C22×D3—361no—
253C22×D3—361no—
254C3×D4—181no—
255C3×D4—181no—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—181no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262#67—362no—
263#67—362no—
264C2×Dic3—61no—
265C2×Dic3D3611yescharacteristic
266C2×Dic3D3611yescharacteristic
267D12—61no—
268D12—61no—
269D12—92no—
270D12—92no—
271D12—182no—
272D12—182no—
273D12—181no—
274D12—181no—
275D12—181no—
276D12—181no—
277D12—181no—
278D12—181no—
279D12—361no—
280D12—361no—
281D12—362no—
282D12—362no—
283C4×D3—362no—
284C4×D3—362no—
285C3×C8—22no—
286C3×C8—21no—
287C3×C8—21no—
288C3×C8—61no—
289C3⋊C8—21no—
290C2×C32#89511yescharacteristic
291C2×C32C2×D2411yescharacteristic
292C2×C32#90811yescharacteristic
293C3⋊D3—122no—
294C3⋊D3—241no—
295C3×D3—241no—
296C2×C9—21no—
297C2×C9—21no—
298C2×C9—21no—
299C2×C9#88411yescharacteristic
300C2×C9D3×D811yescharacteristic
301C2×C9#91411yescharacteristic
302D9—81no—
303D9—122no—
304D9—242no—
305D9—242no—
306D9—241no—
307C2×D4—91no—
308C2×D4—271no—
309C2×D4—271no—
310C2×D4—541no—
311C2×D4—541no—
312C22×C4—541no—
313D8—542no—
314D8—542no—
315C2×C8—31no—
316C2×C8D3×D911yescharacteristic
317C4⋊C4—91no—
318C22⋊C4—541no—
319C42—31no—
320C22×C3—21no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3C2×D3611yescharacteristic
325C22×C3D3×D1211yescharacteristic
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—182no—
330C2×D3—182no—
331C2×D3—362no—
332C2×D3—361no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—361no—
336C2×D3—362no—
337C2×D3—361no—
338C2×D3—361no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—721no—
344C2×D3—721no—
345C2×D3—721no—
346C2×D3—721no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—32no—
350C3×C4—61no—
351C3×C4—361no—
352C3×C4D4×D911yescharacteristic
353C3×C4#370111yescharacteristic
354C3×C4#379911yescharacteristic
355C3×C4#381111yescharacteristic
356Dic3—21no—
357Dic3—121no—
358Dic3D7212yescomplemented
359C32#484311yescharacteristic
360C9—21no—
361C9#478611yescharacteristic, complemented
362C23—181no—
363C23—541no—
364C23—541no—
365D4—181no—
366D4—181no—
367D4—272no—
368D4—541no—
369D4—541no—
370D4—541no—
371D4—1082no—
372D4—1082no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—181no—
376C2×C4—1082no—
377C2×C4C2×D3×D911yescharacteristic
378C8—21no—
379C8—61no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—721no—
384C2×C3#6345711yescharacteristic
385C2×C3C2×D7211yescharacteristic
386C2×C3#6347011yescharacteristic
387C2×C3D3×D2411yescharacteristic
388C2×C3#6379411yescharacteristic
389C2×C3#6391011yescharacteristic
390D3—241no—
391D3—362no—
392D3—362no—
393D3—722no—
394D3—722no—
395D3—721no—
396D3—721no—
397C22—361no—
398C22—361no—
399C22—361no—
400C22—542no—
401C22—1081no—
402C22—1081no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22D3×D3611yescharacteristic
407C4—32no—
408C4—61no—
409C4—361no—
410C4#8863511yescharacteristic
411C4#8866811yescharacteristic
412C3—21no—
413C3#1058852711yescharacteristic, complemented
414C3#1059076511yescharacteristic
415C2—721no—
416C2—1082no—
417C2—2161no—
418C2D3×D7211yescharacteristic
419C2#1172080411yescharacteristic
420C2#1172100511yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
hypercenterC2×C8D3×D9
hypocenterC3×C9#459
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×#88525C2semidirect
2. #11721006C2semidirect
3. #11720993C2semidirect
4. C2×#88417C2semidirect
5. C3×#63380C2semidirect
6. C9×Dic3⋊C8C2semidirect
7. #11720284C2semidirect
37. #63380D3semidirect
45. C4⋊(C3⋊D9)D4semidirect
101. C9×Dic3D8semidirect
116. Dic3⋊C8D9semidirect
245. C3×C9#459semidirect
358. Dic3D72semidirect
361. C9#4786semidirect
413. C3#10588527semidirect

Maximal subgroups

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

subgroupindex
C2×#885252
#117210062
#117209932
C2×#884172
C3×#633802
C9×Dic3⋊C82
#117202842
#105907653
#105885273