Finite Group Databasebeta

← #13159441

Subgroups — #13159441

Order 1728.

1148 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 210 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×#89022C211yesmaximal, characteristic
2C32×#892C211yesmaximal, characteristic, complemented
3C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722179C211yesmaximal, characteristic, complemented
5C3×#63637C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592701—31nomaximal
9#10590610—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6291—21no—
14C2×#6289—21no—
15#89022—22no—
16#89022C414yescomplemented
17C33×C4⋊C4C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×(C2×D3)⋊C4—21no—
21C32×C4⋊Dic3C2211yescharacteristic
22C32×C4×Dic3—21no—
23D3×C3⋊(C3⋊C8)—22no—
24C2×#3828—31no—
25C2×#3799—34no—
26C32×#117—31no—
27C22×C3⋊(C3⋊C8)—31no—
28C3×#892D314yescomplemented
29C2×D3×C3⋊C8—34no—
30#63637—31no—
31#63637D311yescharacteristic, complemented
32#63636—31no—
33#63636—31no—
34#63636—34no—
35#63636—64no—
36C3×#810—34no—
37#63565—34no—
38#63564—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yes—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634C2×C412yes—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3C2×C412yes—
51C32×C4×D3D412yes—
52#6291—24no—
53#6289—24no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5002—94no—
57#4804—91no—
58C22×C32×C4—31no—
59C2×#638—61no—
60#3828—34no—
61#3828—62no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×C4×C3⋊D3—61no—
66C2×#634C2×D311yescharacteristic
67C2×C3×C4×D3C2×D314yes—
68C2×C3×Dic6—34no—
69C2×#626—64no—
70C2×#624—64no—
71#3799—316no—
72#3799—68no—
73C32×C4⋊C4—24no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4C2×D314yes—
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C22⋊C4—61no—
78C32×C42—61no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—34no—
81C2×C3⋊(C3⋊C8)—62no—
82C2×C3⋊(C3⋊C8)—64no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×(C2×D3)⋊C4—24no—
85C3×C4⋊Dic3C2×D314yes—
86C3×C4×Dic3—24no—
87C2×C3×C3⋊C8—34no—
88D3×C3⋊C8—68no—
89C22×C33C2×D411yescharacteristic
90C2×C3⋊(C3⋊D3)—22no—
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C32×D3—21no—
94C2×C32×D3—21no—
95C2×C32×D3C22⋊C412yes—
96C33×C4—41no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—22no—
100C32×Dic3—41no—
101C32×Dic3C22⋊C412yes—
102C2×#239—91no—
103C2×#235—94no—
104C3×#117—34no—
105C22×C3⋊C8—94no—
106C2×D3×C8—91no—
107#892C3⋊D311yescharacteristic, complemented
108C3×#103—91no—
109#835—91no—
110#821—91no—
111#810—34no—
112#810—94no—
113#809—94no—
114#809—94no—
115#809—91no—
116#809—184no—
117C23×C32—31no—
118C22×C3⋊D3—61no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—124no—
121C22×C3×D3#6714yes—
122C2×C32×C4—21no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—31no—
126C2×C32×C4—32no—
127C2×C32×C4—32no—
128C2×C32×C4—44no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132#638—64no—
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—32no—
135C4⋊(C3⋊D3)—38no—
136C4⋊(C3⋊D3)—68no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C4×C3⋊D3—64no—
141#634—21no—
142#634C4×D312yes—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3#6714yes—
146C3×C4×D3C4×D318yes—
147C3×C4×D3D1218yes—
148C3×Dic6—38no—
149C3×Dic6—64no—
150#626—616no—
151#624—616no—
152C3⋊(C3⋊C8)—21no—
153C3⋊(C3⋊C8)—61no—
154C3⋊(C3⋊C8)—64no—
155C3⋊(C3⋊C8)—124no—
156C3×C3⋊C8—64no—
157#416—271noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33#13511yescharacteristic
161C3⋊(C3⋊D3)—42no—
162C32×D3—22no—
163C2×C3×Q8—91no—
164C22×C3×C4—34no—
165C2×#67—184no—
166#239—94no—
167#239—182no—
168#235—916no—
169#235—188no—
170C2×D12—94no—
171C2×D12—91no—
172C2×D12—184no—
173C2×D12—181no—
174C2×C4×D3—184no—
175C2×C4×D3—181no—
176C2×C4×D3C2×C3⋊D311yescharacteristic
177C2×Dic6—34no—
178C2×C3×C8—91no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—34no—
181C3×C4⋊C4D3214yes—
182C3×C4⋊C4C2×C3⋊D311yescharacteristic
183C3×C22⋊C4—64no—
184C3×C42—64no—
185(C2×D3)⋊C4—21no—
186C4⋊Dic3C2×C3⋊D311yescharacteristic
187C4×Dic3—21no—
188C2×C3⋊C8—34no—
189C2×C3⋊C8—94no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—184no—
192C2×C3⋊C8—188no—
193D3×C8—182no—
194C22×C32—24no—
195C22×C32—32no—
196C22×C32—61no—
197C22×C32—61no—
198C22×C32D3×D411yescharacteristic
199C22×C32C2×#6714yes—
200C2×C3⋊D3—62no—
201C2×C3⋊D3—68no—
202C2×C3⋊D3—128no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—244no—
208C2×C3⋊D3—244no—
209C2×C3×D3—24no—
210C2×C3×D3—24no—
211C2×C3×D3(C2×D3)⋊C418yes—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—32no—
215C32×C4—41no—
216C32×C4—44no—
217C32×C4—84no—
218C32×C4—121no—
219C32×C4C2×C4×D314yes—
220C32×C4C2×C4×D311yescharacteristic
221C32×C4C2×D1214yes—
222C32×C4D3×D411yescharacteristic
223C3⋊Dic3—22no—
224C3×Dic3—44no—
225C3×Dic3—68no—
226C3×Dic3(C2×D3)⋊C418yes—
227C2×#41—271no—
228C22×C8—271no—
229#117—31no—
230#103—91no—
231#103—271no—
232#102—271no—
233#102—271no—
234C33#41611yescharacteristic, complemented, Sylow
235C23×C3—34no—
236C22×D3—184no—
237C22×D3—181no—
238C22×D3—364no—
239C22×D3#63811yescharacteristic
240C3×Q8—92no—
241C3×Q8—181no—
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—34no—
246C2×C3×C4—38no—
247C2×C3×C4—38no—
248C2×C3×C4—44no—
249C2×C3×C4—64no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—1816no—
254C2×Dic3—21no—
255C2×Dic3—64no—
256C2×Dic3#63811yescharacteristic
257D12—98no—
258D12—92no—
259D12—188no—
260D12—184no—
261D12—184no—
262D12—181no—
263D12—364no—
264C4×D3—1816no—
265C4×D3—184no—
266C4×D3C4×C3⋊D312yes—
267C4×D3C4⋊(C3⋊D3)12yes—
268Dic6—38no—
269Dic6—64no—
270C3×C8—181no—
271C3⋊C8—64no—
272C3⋊C8—184no—
273C3⋊C8—181no—
274C3⋊C8—364no—
275C2×C32—24no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—62no—
279C2×C32D3×C22⋊C411yescharacteristic
280C2×C32#91611yescharacteristic
281C2×C32#91911yescharacteristic
282C2×C32C2×(C2×D3)⋊C414yes—
283C2×C32#95014yes—
284C2×C32#95014yes—
285C3⋊D3—122no—
286C3⋊D3—128no—
287C3⋊D3—248no—
288C3×D3—28no—
289#41—274no—
290#41—542no—
291C2×Q8—91no—
292C2×D4—271no—
293C2×D4—541no—
294C22×C4—31no—
295C22×C4—541no—
296C2×C8—91no—
297C2×C8—271no—
298C2×C8—542no—
299C4⋊C4—31no—
300C4⋊C4D3×C3⋊D311yescharacteristic
301C22⋊C4—61no—
302C42—61no—
303C22×C3—24no—
304C22×C3—38no—
305C22×C3—64no—
306C22×C3—64no—
307C22×C3D3×#6714yes—
308C22×C3C2×#63811yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—182no—
312C2×D3—188no—
313C2×D3—368no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—364no—
317C2×D3—364no—
318C2×D3—724no—
319C2×D3—724no—
320C2×D3#375212yes—
321C3×C4—24no—
322C3×C4—24no—
323C3×C4—38no—
324C3×C4—41no—
325C3×C4—44no—
326C3×C4—84no—
327C3×C4—124no—
328C3×C4—182no—
329C3×C4C4×D3214yes—
330C3×C4D3×D1214yes—
331C3×C4C2×C4×C3⋊D311yescharacteristic
332C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
333Dic3—41no—
334Dic3—68no—
335Dic3#375212yes—
336C32—24no—
337C32#480411yescharacteristic, complemented
338C32#500214yescomplemented
339C23—31no—
340C23—541no—
341Q8—92no—
342Q8—181no—
343D4—272no—
344D4—544no—
345D4—541no—
346C2×C4—21no—
347C2×C4—31no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—61no—
351C2×C4—181no—
352C2×C4—544no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—181no—
355C8—541no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—68no—
360C2×C3D3×(C2×D3)⋊C414yes—
361C2×C3#6393514yes—
362C2×C3#6394914yes—
363C2×C3C2×#375211yescharacteristic
364C2×C3#6415911yescharacteristic
365C2×C3#6415911yescharacteristic
366D3—22no—
367D3—368no—
368D3—362no—
369D3—728no—
370C22—32no—
371C22—61no—
372C22—61no—
373C22—542no—
374C22—1081no—
375C22—1081no—
376C22D3×#63811yescharacteristic
377C4—32no—
378C4—41no—
379C4—121no—
380C4—182no—
381C4C4×D3×C3⋊D311yescharacteristic
382C4D3×C4⋊(C3⋊D3)11yescharacteristic
383C3—24no—
384C3#1059061014yescomplemented
385C3#1059270111yescharacteristic, complemented
386C2—62no—
387C2—1082no—
388C2D3×#375211yescharacteristic
389C2#1172355811yescharacteristic
390C2#1172358211yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89022C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C32×#892C2semidirect
3. C2×D3×C3⋊(C3⋊C8)C2semidirect
4. #11722179C2semidirect
5. C3×#63637C2semidirect
6. #11722039C2semidirect
7. #11722035C2semidirect
16. #89022C4semidirect
28. C3×#892D3semidirect
31. #63637D3semidirect
107. #892C3⋊D3semidirect
234. C33#416semidirect
337. C32#4804semidirect
338. C32#5002semidirect
384. C3#10590610semidirect
385. C3#10592701semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C32×#8922
C2×D3×C3⋊(C3⋊C8)2
#117221792
C3×#636372
#117220392
#117220352
#105927013
#105906103