Finite Group Databasebeta

← #13138706

Subgroups — #13138706

Order 1728.

550 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 101 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×#88634C211yesmaximal, characteristic
2C2×C8×C3⋊D9C211yesmaximal, characteristic, complemented
3#11720922C211yesmaximal, characteristic, complemented
4C3×#63380C211yesmaximal, characteristic, complemented
5C9×#816C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590956—31nomaximal
9#10588484—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×#6192—21no—
14C2×Dic3×D9—21no—
15#88634—22no—
16#88634C414yescomplemented
17C2×C3×C8×C9C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C8×C3⋊D9—22no—
20C9×C4⋊Dic3C2211yescharacteristic
21C3×C4⋊Dic9C2211yescharacteristic
22#88445—21no—
23Dic3×Dic9—21no—
24C2×#3796—31no—
25C2×C8×C3⋊D3—31no—
26#63849—31no—
27C2×#3701—31no—
28C3×#821—31no—
29C3×#816D311yescharacteristic
30#63565—31no—
31#63562—31no—
32C2×C8×D9—31no—
33C2×C8×D9—61no—
34#63454—31no—
35C9×#103—31no—
36#63395—31no—
37#63380D311yescharacteristic, complemented
38#63369—31no—
39C22×C3⋊D9D411yescharacteristic
40C22×C3×D9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C4×C3⋊D9C2×C412yes—
44C4×C3⋊D9D412yes—
45C2×C9×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C3×D36—21no—
49C3×D36C2×C412yes—
50C9×Dic6—21no—
51C9×Dic6C2×C412yes—
52#6192—24no—
53Dic3×D9—24no—
54C3×C8×C9—21no—
55C3⋊(C9⋊C8)—21no—
56#4804—91no—
57#4772—91no—
58C2×C4×C3⋊D3—31no—
59C2×C3×D12—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#626—61no—
62C2×D3×Dic3—61no—
63#3796—34no—
64#3796—62no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C8×C3⋊D3—62no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3C2×D311yescharacteristic
70#3722—61no—
71Dic32—61no—
72C2×Q8×C9—31no—
73#3701—34no—
74#3701—62no—
75C2×D36—61no—
76C2×D36C2×D311yescharacteristic
77C2×C4×D9—31no—
78C2×C4×D9—61no—
79C2×C4×D9—61no—
80C2×C8×C9—21no—
81C2×C8×C9C2×D311yescharacteristic
82C9×C4⋊C4—31no—
83(C2×D9)⋊C4—61no—
84C4⋊Dic9C2×D311yescharacteristic
85C4×Dic9—61no—
86C2×C9⋊C8—31no—
87C2×C9⋊C8—61no—
88C8×D9—62no—
89C8×D9—64no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—21no—
92C2×C3⋊D9—21no—
93C2×C3⋊D9C22⋊C412yes—
94C2×C3×D9—22no—
95C2×C3×D9—41no—
96C2×C3×D9—41no—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9C22⋊C412yes—
100C9×Dic3—22no—
101C9×Dic3—41no—
102C3×Dic9—41no—
103C2×#239—91no—
104C2×#237—91no—
105C2×D3×C8—91no—
106C2×D3×C8—91no—
107C2×D3×C8—181no—
108#892—91no—
109#892—91no—
110C3×#103—31no—
111C3×#102—91no—
112#835—91no—
113#832—91no—
114#821—31no—
115#816D911yescharacteristic, complemented
116#810—91no—
117#809—91no—
118C22×C3⋊D3—31no—
119C22×C3×D3—61no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—31no—
122C4×C3⋊D3—32no—
123C4×C3⋊D3—32no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—61no—
127C3×D12—32no—
128C3×D12—61no—
129C3×Dic6—21no—
130C3×Dic6C4×D312yes—
131#626—64no—
132D3×Dic3—64no—
133C22×D9—21no—
134C22×D9—31no—
135C22×D9—61no—
136C32×C8—21no—
137C3⋊(C3⋊C8)—61no—
138Q8×C9—32no—
139Q8×C9—61no—
140C2×C4×C9—21no—
141C2×C4×C9—61no—
142C2×C4×C9—61no—
143C2×C4×C9C22×D311yescharacteristic
144C2×Dic9—21no—
145C2×Dic9—31no—
146C2×Dic9—61no—
147D36—21no—
148D36—64no—
149D36C4×D312yes—
150C4×D9—32no—
151C4×D9—32no—
152C4×D9—62no—
153C4×D9—62no—
154C4×D9—64no—
155C8×C9—22no—
156C8×C9—21no—
157C9⋊C8—62no—
158C9⋊C8—61no—
159#416—271noSylow
160C2×C3×C9C2×C22⋊C411yescharacteristic
161C2×C3×C9#13511yescharacteristic
162C2×C3×C9#13511yescharacteristic
163C3⋊D9—22no—
164C3×D9—42no—
165C2×C3×Q8—31no—
166C2×C3×D4—91no—
167C2×#67—181no—
168C22×Dic3—181no—
169#239—94no—
170#239—182no—
171#237—94no—
172#237—182no—
173C2×D12—31no—
174C2×D12—181no—
175C2×C4×D3—91no—
176C2×C4×D3—91no—
177C2×C4×D3—181no—
178C2×C4×D3—181no—
179C2×Dic6C2×D911yescharacteristic
180C2×C3×C8—21no—
181C2×C3×C8C2×D911yescharacteristic
182C2×C3×C8D3211yescharacteristic
183C3×C4⋊C4—31no—
184C3×C4⋊C4—91no—
185(C2×D3)⋊C4—181no—
186(C2×D3)⋊C4—181no—
187C4⋊Dic3—31no—
188C4⋊Dic3C2×D911yescharacteristic
189C4×Dic3—181no—
190C4×Dic3—181no—
191C2×C3⋊C8—91no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—181no—
194D3×C8—182no—
195D3×C8—182no—
196D3×C8—184no—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—32no—
199C2×C3⋊D3—61no—
200C2×C3⋊D3—61no—
201C2×C3×D3—62no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4D3×D411yescharacteristic
206C3⋊Dic3—32no—
207C3×Dic3—22no—
208C3×Dic3—41no—
209C3×Dic3—121no—
210C22×C9—21no—
211C22×C9D3×D411yescharacteristic
212C2×D9—22no—
213C2×D9—32no—
214C2×D9—41no—
215C2×D9—41no—
216C2×D9—62no—
217C2×D9—62no—
218C2×D9—62no—
219C2×D9—61no—
220C2×D9—61no—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—62no—
224C4×C9—121no—
225C4×C9C2×C4×D311yescharacteristic
226C4×C9D3×D411yescharacteristic
227Dic9—32no—
228Dic9—41no—
229Dic9—62no—
230C2×#41—271no—
231C22×C8—271no—
232#117—271no—
233#103—31no—
234#103—271no—
235#102—91no—
236#102—271no—
237C3×C9#41611yescharacteristic, complemented, Sylow
238C23×C3—181no—
239C22×D3—61no—
240C22×D3—91no—
241C22×D3—91no—
242C22×D3—181no—
243C3×Q8—32no—
244C3×Q8—61no—
245C3×D4—92no—
246C3×D4—181no—
247C2×C3×C4—21no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—181no—
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253#67—184no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—61no—
257C2×Dic3—91no—
258C2×Dic3—91no—
259C2×Dic3—181no—
260C2×Dic3—184no—
261D12—32no—
262D12—61no—
263D12—184no—
264C4×D3—92no—
265C4×D3—92no—
266C4×D3—92no—
267C4×D3—92no—
268C4×D3—182no—
269C4×D3—182no—
270C4×D3—184no—
271Dic6—21no—
272Dic6C4×D912yes—
273C3×C8—22no—
274C3×C8—21no—
275C3×C8—21no—
276C3⋊C8—182no—
277C3⋊C8—181no—
278C3⋊C8—181no—
279C2×C32D3×C22⋊C411yescharacteristic
280C2×C32#91611yescharacteristic
281C2×C32#91911yescharacteristic
282C3⋊D3—62no—
283C3×D3—122no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9D3×C22⋊C411yescharacteristic
288C2×C9#91211yescharacteristic
289C2×C9#91611yescharacteristic
290D9—42no—
291D9—64no—
292D9—62no—
293#41—274no—
294#41—542no—
295C2×Q8—31no—
296C2×D4—91no—
297C2×D4—541no—
298C22×C4—271no—
299C22×C4—541no—
300C2×C8—271no—
301C2×C8—542no—
302C2×C8D3×D911yescharacteristic
303C4⋊C4—31no—
304C4⋊C4—91no—
305C22⋊C4—541no—
306C42—541no—
307C22×C3—21no—
308C22×C3—182no—
309C22×C3—361no—
310C22×C3—361no—
311C22×C3D4×D911yescharacteristic
312C22×C3#380211yescharacteristic
313C2×D3—62no—
314C2×D3—92no—
315C2×D3—92no—
316C2×D3—121no—
317C2×D3—121no—
318C2×D3—182no—
319C2×D3—182no—
320C2×D3—182no—
321C2×D3—181no—
322C2×D3—181no—
323C2×D3—181no—
324C2×D3—181no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—62no—
328C3×C4—121no—
329C3×C4—361no—
330C3×C4C2×C4×D911yescharacteristic
331C3×C4D4×D911yescharacteristic
332C3×C4C4×D3211yescharacteristic
333C3×C4#381111yescharacteristic
334Dic3—22no—
335Dic3—41no—
336Dic3—92no—
337Dic3—92no—
338Dic3—121no—
339Dic3—182no—
340C32#480411yescharacteristic
341C9—21no—
342C9#477211yescharacteristic, complemented
343C23—181no—
344C23—271no—
345Q8—32no—
346Q8—61no—
347D4—92no—
348D4—181no—
349D4—544no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—181no—
353C2×C4—271no—
354C2×C4—272no—
355C2×C4—272no—
356C2×C4—544no—
357C2×C4C2×D3×D911yescharacteristic
358C8—21no—
359C8—541no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—362no—
364C2×C3C22⋊C4×D911yescharacteristic
365C2×C3#6347811yescharacteristic
366C2×C3#6348111yescharacteristic
367C2×C3#6380911yescharacteristic
368C2×C3#6381011yescharacteristic
369C2×C3#6392111yescharacteristic
370D3—122no—
371D3—184no—
372D3—182no—
373D3—182no—
374C22—182no—
375C22—272no—
376C22—361no—
377C22—361no—
378C22—541no—
379C22—541no—
380C22#8864611yescharacteristic
381C4—62no—
382C4—121no—
383C4—272no—
384C4—361no—
385C4C4×D3×D911yescharacteristic
386C4#8866811yescharacteristic
387C3—21no—
388C3#1058848411yescharacteristic, complemented
389C3#1059095611yescharacteristic
390C2—362no—
391C2—542no—
392C2#1172082311yescharacteristic
393C2#1172083111yescharacteristic
394C2#1172104811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88634C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#416
LCSG ▷ 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×C8×C3⋊D9C2semidirect
3. #11720922C2semidirect
4. C3×#63380C2semidirect
5. C9×#816C2semidirect
6. #11720297C2semidirect
7. #11720266C2semidirect
16. #88634C4semidirect
37. #63380D3semidirect
115. #816D9semidirect
237. C3×C9#416semidirect
342. C9#4772semidirect
388. C3#10588484semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C2×C8×C3⋊D92
#117209222
C3×#633802
C9×#8162
#117202972
#117202662
#105909563
#105884843