Finite Group Databasebeta

← #13137853

Subgroups — #13137853

Order 1728.

536 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
3C2×#88413C211yesmaximal, characteristic, complemented
4#11720528C211yesmaximal, characteristic, complemented
5C9×#810C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720284C211yesmaximal, characteristic, complemented
8#10590609—31nomaximal
9#10588483—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×C4×D9—21no—
12C2×C3×C4×D9C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×#6192—21no—
15C2×(C3⋊D9)⋊C4—21no—
16#88635—22no—
17#88635C414yescomplemented
18C3×C9×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C3×C4⋊Dic9C2211yescharacteristic
21C3×Dic9⋊C4—21no—
22C2×C9×C3⋊C8C2211yescharacteristic
23#88413—22no—
24C2×#3799—31no—
25C3×D3×C4⋊C4—31no—
26C2×#3711—31no—
27C2×#3701—31no—
28#63636—31no—
29C3×#810D311yescharacteristic
30#63565—31no—
31#63564—31no—
32C2×#3663—31no—
33C4⋊C4×D9D311yescharacteristic, complemented
34C9×#103—31no—
35#63395—31no—
36#63380—31no—
37#63369—31no—
38#63369—61no—
39C22×C3⋊D9—21no—
40C22×C3×D9D411yescharacteristic
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)C2×C412yescomplemented
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×C4×D9—42no—
49C3×C4×D9C2×C412yes—
50C3×C4×D9D412yes—
51C9×Dic6—21no—
52C9×Dic6C2×C412yescomplemented
53#6192—24no—
54(C3⋊D9)⋊C4—24no—
55C3⋊(C9⋊C8)—21no—
56C9×C3⋊C8—21no—
57#4966—91no—
58#4803—91no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—61no—
62C2×C3×Dic6C2×D311yescharacteristic
63C2×#626—61no—
64C2×#624—61no—
65#3799—34no—
66#3799—62no—
67C32×C4⋊C4C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C3×C4⋊Dic3—31no—
70C3×Dic3⋊C4—61no—
71C2×C3×C3⋊C8C2×D311yescharacteristic
72#3711—62no—
73C2×Q8×C9—31no—
74#3701—34no—
75#3701—62no—
76C2×D36—31no—
77C2×D36—61no—
78C2×D36—61no—
79C2×C4×D9—21no—
80C2×C4×D9—61no—
81C2×C4×D9C2×D311yescharacteristic
82C2×C8×C9—31no—
83C9×C4⋊C4—21no—
84C9×C4⋊C4C2×D311yescharacteristic
85C4⋊Dic9C2×D311yescharacteristic
86Dic9⋊C4—21no—
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89#3663—62no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—22no—
92C2×C3⋊D9—41no—
93C2×C3⋊D9—41no—
94C2×C3×D9—21no—
95C2×C3×D9—21no—
96C2×C3×D9C22⋊C412yes—
97C3×C4×C9—41no—
98C3×C4×C9C22×C411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C9×Dic3—22no—
101C3×Dic9—41no—
102C3×Dic9C22⋊C412yes—
103C2×#239—91no—
104C2×#235—91no—
105C2×C3×C4⋊C4—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—91no—
108D3×C4⋊C4—31no—
109C3×#103—31no—
110#835—91no—
111#821—91no—
112#810—91no—
113#810D911yescharacteristic, complemented
114#809—91no—
115#809—91no—
116#809—91no—
117#809—181no—
118C22×C3⋊D3—61no—
119C22×C3×D3—31no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C4⋊(C3⋊D3)—32no—
123C4⋊(C3⋊D3)—61no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C3×C4×D3—32no—
128C3×C4×D3—32no—
129C3×C4×D3—122no—
130C3×Dic6—21no—
131C3×Dic6C4×D312yes—
132#626—64no—
133#624—64no—
134C22×D9—61no—
135C22×D9—121no—
136C22×D9#6711yescharacteristic
137C3⋊(C3⋊C8)—61no—
138C3×C3⋊C8—21no—
139Q8×C9—32no—
140Q8×C9—61no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—41no—
144C2×C4×C9—61no—
145C2×C4×C9C22×D311yescharacteristic
146C2×Dic9—21no—
147C2×Dic9#6711yescharacteristic
148D36—32no—
149D36—62no—
150D36—64no—
151D36—61no—
152D36—121no—
153C4×D9—42no—
154C4×D9—64no—
155C4×D9C4×D312yes—
156C4×D9D1212yes—
157C8×C9—61no—
158C9⋊C8—61no—
159C9⋊C8—121no—
160#417—271noSylow
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
163C2×C3×C9#13711yescharacteristic
164C3⋊D9—42no—
165C3×D9—22no—
166C2×C3×Q8—31no—
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169C2×#67—181no—
170#239—94no—
171#239—182no—
172#235—94no—
173#235—182no—
174C2×D12—91no—
175C2×D12—91no—
176C2×D12—181no—
177C2×D12—181no—
178C2×C4×D3—31no—
179C2×C4×D3—61no—
180C2×C4×D3—181no—
181C2×C4×D3—181no—
182C2×Dic6C2×D911yescharacteristic
183C2×C3×C8—31no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—91no—
186C3×C4⋊C4—181no—
187C3×C4⋊C4C2×D911yescharacteristic
188C3×C4⋊C4D3211yescharacteristic
189C4⋊Dic3—31no—
190Dic3⋊C4—61no—
191#208—182no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—181no—
195C2×C3⋊C8C2×D911yescharacteristic
196ΓL(1,𝔽25)—182no—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—62no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C2×C3×D3—32no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C32×C4—41no—
205C32×C4C2×C4×D311yescharacteristic
206C32×C4D3×D411yescharacteristic
207C3×Dic3—22no—
208C3×Dic3—32no—
209C3×Dic3—121no—
210C22×C9—21no—
211C22×C9C2×#6711yescharacteristic
212C2×D9—21no—
213C2×D9—21no—
214C2×D9—62no—
215C2×D9—122no—
216C2×D9—121no—
217C2×D9—121no—
218C2×D9—241no—
219C2×D9—241no—
220C2×D9(C2×D3)⋊C412yes—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—41no—
224C4×C9—62no—
225C4×C9—81no—
226C4×C9C2×C4×D311yescharacteristic
227C4×C9C2×D1211yescharacteristic
228Dic9—41no—
229Dic9(C2×D3)⋊C412yes—
230C2×#41—271no—
231C2×M16—271no—
232C2×C4⋊C4—91no—
233#103—31no—
234#103—271no—
235#102—271no—
236#102—271no—
237C3×C9#41711yescharacteristic, complemented, Sylow
238C23×C3—91no—
239C22×D3—31no—
240C22×D3—181no—
241C22×D3—181no—
242C22×D3—361no—
243C3×Q8—32no—
244C3×Q8—61no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—61no—
250C2×C3×C4—91no—
251C2×C3×C4—92no—
252C2×C3×C4—92no—
253C2×C3×C4—181no—
254C2×C3×C4—362no—
255C2×C3×C4C22×D911yescharacteristic
256C2×C3×C4C2×D3211yescharacteristic
257#67—184no—
258C2×Dic3—21no—
259C2×Dic3—31no—
260C2×Dic3—61no—
261D12—92no—
262D12—92no—
263D12—182no—
264D12—184no—
265D12—181no—
266D12—181no—
267D12—361no—
268C4×D3—32no—
269C4×D3—32no—
270C4×D3—122no—
271C4×D3—184no—
272C4×D3—184no—
273Dic6—21no—
274Dic6C4×D912yescomplemented
275C3×C8—61no—
276C3⋊C8—21no—
277C3⋊C8—181no—
278C3⋊C8—181no—
279C3⋊C8—361no—
280C2×C32D3×C22⋊C411yescharacteristic
281C2×C32#91411yescharacteristic
282C2×C32#91811yescharacteristic
283C3⋊D3—122no—
284C3×D3—62no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9C2×(C2×D3)⋊C411yescharacteristic
289C2×C9#93211yescharacteristic
290C2×C9#94211yescharacteristic
291D9—22no—
292D9—122no—
293D9—242no—
294#41—274no—
295#41—542no—
296C2×Q8—31no—
297C2×D4—271no—
298C2×D4—541no—
299C22×C4—91no—
300C22×C4—181no—
301C22×C4—541no—
302M16—542no—
303C2×C8—31no—
304C2×C8—271no—
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4D3×D911yescharacteristic
308C22×C3—21no—
309C22×C3—92no—
310C22×C3—181no—
311C22×C3—181no—
312C22×C3D4×D911yescharacteristic
313C22×C3D3×#6711yescharacteristic
314C2×D3—32no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—182no—
318C2×D3—182no—
319C2×D3—362no—
320C2×D3—361no—
321C2×D3—361no—
322C2×D3—361no—
323C2×D3—361no—
324C2×D3—721no—
325C2×D3—721no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—62no—
331C3×C4—81no—
332C3×C4—92no—
333C3×C4—361no—
334C3×C4C2×C4×D911yescharacteristic
335C3×C4D4×D911yescharacteristic
336C3×C4C4×D3211yescharacteristic
337C3×C4D3×D1211yescharacteristic
338Dic3—22no—
339Dic3—32no—
340Dic3—121no—
341C32#480311yescharacteristic
342C9—21no—
343C9#496611yescharacteristic, complemented
344C23—91no—
345C23—541no—
346Q8—32no—
347Q8—61no—
348D4—272no—
349D4—544no—
350D4—541no—
351C2×C4—21no—
352C2×C4—61no—
353C2×C4—91no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—181no—
357C2×C4—362no—
358C2×C4—544no—
359C2×C4C2×D3×D911yescharacteristic
360C8—61no—
361C8—541no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—182no—
366C2×C3C22⋊C4×D911yescharacteristic
367C2×C3#6347611yescharacteristic
368C2×C3#6348011yescharacteristic
369C2×C3D3×(C2×D3)⋊C411yescharacteristic
370C2×C3#6393411yescharacteristic
371C2×C3#6394811yescharacteristic
372D3—62no—
373D3—362no—
374D3—362no—
375D3—722no—
376C22—92no—
377C22—181no—
378C22—181no—
379C22—542no—
380C22—1081no—
381C22—1081no—
382C22D9×#6711yescharacteristic
383C4—41no—
384C4—62no—
385C4—92no—
386C4—361no—
387C4C4×D3×D911yescharacteristic
388C4D9×D1211yescharacteristic
389C3—21no—
390C3#1058848311yescharacteristic, complemented
391C3#1059060911yescharacteristic
392C2—182no—
393C2—1082no—
394C2D9×(C2×D3)⋊C411yescharacteristic
395C2#1172118411yescharacteristic
396C2#1172122211yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88635C2
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×C4⋊C4×D9C2semidirect
3. C2×#88413C2semidirect
4. #11720528C2semidirect
5. C9×#810C2semidirect
6. #11720297C2semidirect
7. #11720284C2semidirect
17. #88635C4semidirect
33. C4⋊C4×D9D3semidirect
44. C4⋊(C3⋊D9)C2×C4semidirect
52. C9×Dic6C2×C4semidirect
113. #810D9semidirect
237. C3×C9#417semidirect
274. Dic6C4×D9semidirect
343. C9#4966semidirect
390. C3#10588483semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
C3×C4⋊C4×D92
C2×#884132
#117205282
C9×#8102
#117202972
#117202842
#105906093
#105884833