Finite Group Databasebeta

← #13173436

Subgroups — #13173436

Order 1728.

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

normal subgroups only

This group has 748 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
1C3×#41×C3⋊D3C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3#11724511C211yesmaximal, characteristic, complemented
4#11723110C212yesmaximal, complemented
5#11723108C212yesmaximal, complemented
6#10593839—31nomaximal
7C22×C3⋊#157—21no—
8C33×#41C411yescharacteristic, complemented
9C3×#3832C2211yescharacteristic
10C3×Q8×C3⋊D3C411yescharacteristic, complemented
11C3×#3830—21no—
12C3×#3830C2211yescharacteristic
13C3×D4×C3⋊D3—21no—
14C3×D4×C3⋊D3C411yescharacteristic, complemented
15C3×#3828—21no—
16C3×#3828C411yescharacteristic, complemented
17C2×C3×C4×C3⋊D3—21no—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19#88993—21no—
20C4×C3⋊#157—21no—
21C4×C3⋊#157C2212yescomplemented
22#88989C2212yescomplemented
23#88988—21no—
24#41×C3⋊D3D311yescharacteristic
25C3×D3×#41—62no—
26C2×C4×#157—31no—
27#64283—31no—
28#63785—32no—
29#63783—32no—
30C22×C3×C3⋊D3—21no—
31C22×C3×C3⋊D3D411yescharacteristic
32C2×C3⋊#157—24no—
33C2×C3⋊#157—22no—
34C33×Q8C2×C411yescharacteristic, complemented
35C33×D4—21no—
36C33×D4C2×C411yescharacteristic, complemented
37C2×C33×C4—21no—
38C2×C33×C4C2×C411yescharacteristic
39C3×#638—21no—
40C3×#638—21no—
41C3×#638—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3C2×C411yescharacteristic
50C3×C4×C3⋊D3D411yescharacteristic
51C3×C4×C3⋊D3D411yescharacteristic
52C3×C4×C3⋊D3C2311yescharacteristic
53C3×#634—21no—
54C3×#634C2×C411yescharacteristic, complemented
55C3⋊#622—22no—
56C2×#837—91nomaximal
57C22×#157—61no—
58C32×#41—22no—
59C32×#41—42no—
60C32×#41Dic311yescharacteristic, complemented
61#3832C2×D311yescharacteristic
62Q8×C3⋊D3Dic311yescharacteristic, complemented
63#3830—21no—
64#3830C2×D311yescharacteristic
65D4×C3⋊D3—21no—
66D4×C3⋊D3Dic311yescharacteristic, complemented
67#3828—21no—
68#3828Dic311yescharacteristic, complemented
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C3×#239—62no—
72C3×D3×Q8—62no—
73C3×#237—62no—
74C3×#237—64no—
75C3×D3×D4—62no—
76C3×D3×D4—64no—
77C3×#235—62no—
78C3×#235—64no—
79C2×C3×C4×D3—62no—
80C2×C3×C4×D3—64no—
81#3792—61no—
82C4×#157—32no—
83C4×#157—61no—
84#3788—32no—
85#3787—61no—
86C22×C33—21no—
87C22×C33C22⋊C411yescharacteristic
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—42no—
91C2×C3×C3⋊D3C22⋊C411yescharacteristic
92C2×C3×C3⋊D3C2×D411yescharacteristic
93C3⋊#157—24no—
94C33×C4—21no—
95C33×C4C22⋊C411yescharacteristic
96C33×C4C22×C411yescharacteristic
97C3×C3⋊Dic3—21no—
98C3×C3⋊Dic3C22⋊C411yescharacteristic
99C3×C3⋊Dic3C2×D411yescharacteristic
100C2×C3×#41—91no—
101D3×#41—62no—
102C2×C4×Dic3—91no—
103C2×#208—91no—
104#837—92no—
105#837—92no—
106C22×C3⋊D3—21no—
107C22×C3⋊D3#6711yescharacteristic
108C22×C3×D3—62no—
109C22×C3×D3—64no—
110C2×#157—64no—
111C2×#157—62no—
112C32×Q8—22no—
113C32×Q8—42no—
114C32×Q8C2×Dic311yescharacteristic, complemented
115C32×D4—21no—
116C32×D4—24no—
117C32×D4—22no—
118C32×D4—44no—
119C32×D4—42no—
120C32×D4C2×Dic311yescharacteristic, complemented
121C2×C32×C4—21no—
122C2×C32×C4—24no—
123C2×C32×C4—22no—
124C2×C32×C4—44no—
125C2×C32×C4—42no—
126C2×C32×C4C2×Dic311yescharacteristic
127#638—21no—
128#638—21no—
129#638—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3#6711yescharacteristic
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)C2×Dic311yescharacteristic, complemented
134C4×C3⋊D3—21no—
135C4×C3⋊D3—21no—
136C4×C3⋊D3—21no—
137C4×C3⋊D3C2×Dic311yescharacteristic
138C4×C3⋊D3#6711yescharacteristic
139C4×C3⋊D3#6711yescharacteristic
140C4×C3⋊D3C22×D311yescharacteristic
141#634—21no—
142#634C2×Dic311yescharacteristic, complemented
143C3×#67—64no—
144C3×#67—64no—
145C3×#67—64no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—64no—
148C3×D12—62no—
149C3×D12—64no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×C4×D3—64no—
155C3×C4×D3—64no—
156C3×C4×D3—64no—
157C3×Dic6—62no—
158C3×Dic6—64no—
159#622—62no—
160C2×#104—271noSylow
161C2×C33—21no—
162C2×C33—41no—
163C2×C33C2×C22⋊C411yescharacteristic
164C3×C3⋊D3—21no—
165C3×C3⋊D3—41no—
166C3×C3⋊D3#10412yescomplemented
167C3×#41—22no—
168C3×#41—42no—
169C3×#41—91no—
170C3×#41—91no—
171C3×#41—91no—
172C3×#41—181no—
173C3×#41—181no—
174C3×#41#15711yescharacteristic, complemented
175C2×C3×Q8—91no—
176C2×C3×D4—91no—
177C2×C3×D4—181no—
178C22×C3×C4—91no—
179C22×C3×C4—181no—
180C22×Dic3—181no—
181#239—62no—
182D3×Q8—62no—
183#237—62no—
184#237—64no—
185D3×D4—62no—
186D3×D4—64no—
187#235—62no—
188#235—64no—
189C2×C4×D3—62no—
190C2×C4×D3—64no—
191C4×Dic3—92no—
192C4×Dic3—181no—
193#208—92no—
194#208—181no—
195C2×C3⋊C8—181no—
196C22×C32—21no—
197C22×C32—24no—
198C22×C32—22no—
199C22×C32—44no—
200C22×C32—42no—
201C22×C32#21711yescharacteristic
202C2×C3⋊D3—21no—
203C2×C3⋊D3—22no—
204C2×C3⋊D3—42no—
205C2×C3⋊D3#21711yescharacteristic
206C2×C3⋊D3C2×#6711yescharacteristic
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—64no—
210C2×C3×D3—124no—
211C2×C3×D3—128no—
212#157—64no—
213C32×C4—21no—
214C32×C4—24no—
215C32×C4—22no—
216C32×C4—22no—
217C32×C4—44no—
218C32×C4—42no—
219C32×C4—42no—
220C32×C4#21711yescharacteristic
221C32×C4C22×Dic311yescharacteristic
222C3⋊Dic3—21no—
223C3⋊Dic3#21711yescharacteristic
224C3⋊Dic3C2×#6711yescharacteristic
225C3×Dic3—62no—
226C3×Dic3—62no—
227C3×Dic3—64no—
228C2×#41—91no—
229C2×M16—271no—
230C2×C42—271no—
231#104—272no—
232#104—272no—
233C33C2×#10411yescharacteristic, complemented, Sylow
234C23×C3—91no—
235C23×C3—181no—
236C22×D3—62no—
237C22×D3—64no—
238C3×Q8—22no—
239C3×Q8—42no—
240C3×Q8—91no—
241C3×Q8—181no—
242C3×Q8C2×#15711yescharacteristic, complemented
243C3×D4—24no—
244C3×D4—21no—
245C3×D4—22no—
246C3×D4—44no—
247C3×D4—42no—
248C3×D4—91no—
249C3×D4—181no—
250C3×D4—181no—
251C3×D4—181no—
252C3×D4—181no—
253C3×D4C2×#15711yescharacteristic, complemented
254C2×C3×C4—24no—
255C2×C3×C4—21no—
256C2×C3×C4—22no—
257C2×C3×C4—44no—
258C2×C3×C4—42no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—91no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×#15711yescharacteristic
269#67—64no—
270#67—64no—
271#67—64no—
272C2×Dic3—62no—
273C2×Dic3—64no—
274C2×Dic3—182no—
275C2×Dic3—184no—
276D12—62no—
277D12—64no—
278C4×D3—62no—
279C4×D3—62no—
280C4×D3—62no—
281C4×D3—62no—
282C4×D3—64no—
283C4×D3—64no—
284C4×D3—64no—
285Dic6—62no—
286Dic6—64no—
287C3⋊C8—182no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—42no—
291C2×C32—42no—
292C2×C32—41no—
293C2×C32—44no—
294C2×C32—84no—
295C2×C32—82no—
296C2×C32C2×#21711yescharacteristic
297C3⋊D3—21no—
298C3⋊D3—41no—
299C3⋊D3#83712yescomplemented
300C3×D3—64no—
301C3×D3—122no—
302C3×D3—124no—
303#41—91no—
304#41—91no—
305#41—91no—
306#41—181no—
307#41—181no—
308#41C3⋊#15711yescharacteristic, complemented
309C2×Q8—91no—
310C2×D4—91no—
311C2×D4—181no—
312C22×C4—91no—
313C22×C4—181no—
314C22×C4—541no—
315M16—272no—
316M16—541no—
317C2×C8—541no—
318C42—272no—
319C42—541no—
320C22×C3—24no—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—44no—
324C22×C3—42no—
325C22×C3—91no—
326C22×C3—91no—
327C22×C3—182no—
328C22×C3—181no—
329C22×C3—362no—
330C22×C3C22⋊#15711yescharacteristic
331C2×D3—62no—
332C2×D3—62no—
333C2×D3—64no—
334C2×D3—124no—
335C2×D3—128no—
336C3×C4—24no—
337C3×C4—21no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—44no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—91no—
344C3×C4—91no—
345C3×C4—181no—
346C3×C4C22⋊#15711yescharacteristic
347C3×C4C22×#15711yescharacteristic
348Dic3—62no—
349Dic3—62no—
350Dic3—64no—
351Dic3—184no—
352C32—22no—
353C32—42no—
354C32C2×#83711yescharacteristic, complemented
355C23—91no—
356C23—181no—
357Q8—91no—
358Q8—181no—
359Q8C2×C3⋊#15711yescharacteristic, complemented
360D4—21no—
361D4—91no—
362D4—181no—
363D4—181no—
364D4—181no—
365D4—181no—
366D4C2×C3⋊#15711yescharacteristic, complemented
367C2×C4—21no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4—542no—
378C2×C4—544no—
379C2×C4C2×C3⋊#15711yescharacteristic
380C8—542no—
381C2×C3—21no—
382C2×C3—22no—
383C2×C3—42no—
384C2×C3—44no—
385C2×C3—41no—
386C2×C3—42no—
387C2×C3—82no—
388C2×C3—84no—
389C2×C3—92no—
390C2×C3—181no—
391C2×C3—361no—
392C2×C3C2×C22⋊#15711yescharacteristic
393D3—64no—
394D3—122no—
395D3—124no—
396C22—21no—
397C22—91no—
398C22—91no—
399C22—182no—
400C22—181no—
401C22—362no—
402C22#8899411yescharacteristic
403C4—21no—
404C4—91no—
405C4—91no—
406C4—181no—
407C4—544no—
408C4#8899411yescharacteristic
409C4C22×C3⋊#15711yescharacteristic
410C3—22no—
411C3—42no—
412C3#1059383911yescharacteristic, complemented
413C2—21no—
414C2—41no—
415C2—92no—
416C2—181no—
417C2—361no—
418C2C2×#8899411yescharacteristic
subgroupG/H
centerC4#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC33×#41C4
socleC2×C33C2×C22⋊C4
prime radicalC3×#41×C3⋊D3C2
cosocleC3×C4×C3⋊D3C23
hypercenter#41C3⋊#157
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#157C2semidirect
3. #11724511C2semidirect
4. #11723110C2semidirect
5. #11723108C2semidirect
8. C33×#41C4semidirect
10. C3×Q8×C3⋊D3C4semidirect
14. C3×D4×C3⋊D3C4semidirect
16. C3×#3828C4semidirect
21. C4×C3⋊#157C22semidirect
22. #88989C22semidirect
34. C33×Q8C2×C4semidirect
36. C33×D4C2×C4semidirect
45. C3×C4⋊(C3⋊D3)C2×C4semidirect
54. C3×#634C2×C4semidirect
60. C32×#41Dic3semidirect
62. Q8×C3⋊D3Dic3semidirect
66. D4×C3⋊D3Dic3semidirect
68. #3828Dic3semidirect
114. C32×Q8C2×Dic3semidirect
120. C32×D4C2×Dic3semidirect
133. C4⋊(C3⋊D3)C2×Dic3semidirect
142. #634C2×Dic3semidirect
166. C3×C3⋊D3#104semidirect
174. C3×#41#157semidirect
233. C33C2×#104semidirect
242. C3×Q8C2×#157semidirect
253. C3×D4C2×#157semidirect
299. C3⋊D3#837semidirect
308. #41C3⋊#157semidirect
354. C32C2×#837semidirect
359. Q8C2×C3⋊#157semidirect
366. D4C2×C3⋊#157semidirect
412. C3#10593839semidirect

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
C2×C4×C3⋊#1572
#117245112
#117231102
#117231082
#105938393
C2×#8379