Finite Group Databasebeta

← #13166380

Subgroups — #13166380

Order 1728.

748 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 78 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⋊D3×#235C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4#11722712C212yesmaximal, complemented
5#11722710C212yesmaximal, complemented
6#10593839—31nomaximal
7C22×C3⋊#157—21no—
8#89076C411yescharacteristic, complemented
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C32×#235C411yescharacteristic, complemented
11C3⋊D3×#67—21no—
12#89028—21no—
13C3⋊D3×D12C411yescharacteristic, complemented
14C4×D3×C3⋊D3—21no—
15#89020—21no—
16#89018C2211yescharacteristic
17C3⋊D3×Dic6C411yescharacteristic, complemented
18#89014C2211yescharacteristic
19C4×C3⋊#157—21no—
20C4×C3⋊#157C2212yescomplemented
21C3×#3792—21no—
22C3×#3788C2212yescomplemented
23C3×#3787—21no—
24#41×C3⋊D3—31no—
25D3×#235—62no—
26C2×C4×#157—31no—
27#64283D311yescharacteristic, complemented
28#63785—32no—
29#63783—32no—
30C22×C3×C3⋊D3D411yescharacteristic
31C2×D3×C3⋊D3—21no—
32C2×C3⋊#157—24no—
33C2×C3⋊#157—22no—
34C2×C33×C4C2×C411yescharacteristic
35#6312—21no—
36C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic, complemented
37C4×C3⋊(C3⋊D3)—21no—
38#6308C2×C411yescharacteristic, complemented
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×C4×C3⋊D3C2×C411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3D411yescharacteristic
43C3×C4×C3⋊D3C2311yescharacteristic
44C32×#67—21no—
45C32×D12C2×C411yescharacteristic, complemented
46C32×C4×D3—21no—
47C32×Dic6C2×C411yescharacteristic, complemented
48#6293—21no—
49#6292—21no—
50#6291—21no—
51#6290—21no—
52#6289—21no—
53Dic3×C3⋊D3—21no—
54D3×C3⋊Dic3—21no—
55C3×#622—22no—
56C2×#825—91nomaximal
57C22×#157—61no—
58C32×#41—31no—
59#3832—31no—
60Q8×C3⋊D3—31no—
61#3830—31no—
62#3830—61no—
63D4×C3⋊D3—31no—
64D4×C3⋊D3—61no—
65#3828—31no—
66#3828—62no—
67#3828—61no—
68#3828—122no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C3×#235—22no—
72C2×C3×C4×D3—62no—
73D3×#67—64no—
74#3804—64no—
75D3×D12—62no—
76C4×D32—64no—
77#3799—62no—
78#3798—64no—
79#3795—62no—
80D3×Dic6—62no—
81#3792—21no—
82C4×#157—32no—
83C4×#157—61no—
84#3788C2×D312yescomplemented
85#3787—21no—
86C22×C33C22⋊C411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3C22⋊C411yescharacteristic
90C2×C3×C3⋊D3C2×D411yescharacteristic
91C2×C32×D3—21no—
92D3×C3⋊D3—42no—
93C3⋊#157—24no—
94C33×C4C22⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C3⋊(C3⋊Dic3)—21no—
97C3×C3⋊Dic3C22⋊C411yescharacteristic
98C3×C3⋊Dic3C2×D411yescharacteristic
99C32×Dic3—21no—
100D3×#41—182no—
101C2×#235—91no—
102C2×C3×M16—91no—
103C2×C4×Dic3—91no—
104#825—92no—
105#825—92no—
106C22×C3⋊D3—61no—
107C22×C3⋊D3D1211yescharacteristic
108C22×C3×D3—62no—
109C2×D32—64no—
110C2×#157—64no—
111C2×#157—62no—
112C32×Q8—31no—
113C32×D4—31no—
114C32×D4—61no—
115C2×C32×C4—22no—
116C2×C32×C4—42no—
117C2×C32×C4—61no—
118C2×C32×C4C4×D311yescharacteristic
119#638—61no—
120#638—61no—
121#638—61no—
122#638—64no—
123#638—124no—
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3#6711yescharacteristic
126C4⋊(C3⋊D3)—31no—
127C4⋊(C3⋊D3)—62no—
128C4⋊(C3⋊D3)—61no—
129C4⋊(C3⋊D3)—122no—
130C4×C3⋊D3—61no—
131C4×C3⋊D3—61no—
132C4×C3⋊D3—61no—
133C4×C3⋊D3—64no—
134C4×C3⋊D3—124no—
135C4×C3⋊D3C4×D311yescharacteristic
136C4×C3⋊D3D1211yescharacteristic
137C4×C3⋊D3#6711yescharacteristic
138C4×C3⋊D3C22×D311yescharacteristic
139#634—31no—
140#634—62no—
141#634—61no—
142#634—122no—
143C3×#67—24no—
144C2×C3×Dic3—62no—
145C3×D12—22no—
146C3×C4×D3—24no—
147C3×C4×D3—62no—
148C3×C4×D3—62no—
149C3×C4×D3—62no—
150C3×C4×D3—62no—
151C3×Dic6—22no—
152#627—64no—
153#626—64no—
154#626—64no—
155#625—64no—
156#624—64no—
157D3×Dic3—64no—
158D3×Dic3—64no—
159#622—22no—
160C2×#104—271noSylow
161C2×C33—21no—
162C2×C33C2×C22⋊C411yescharacteristic
163C3⋊(C3⋊D3)—41no—
164C3×C3⋊D3—21no—
165C3×C3⋊D3#10412yescomplemented
166C32×D3—41no—
167C3×#41—62no—
168C22×C3×C4—91no—
169C2×#67—181no—
170C22×Dic3—181no—
171#239—182no—
172D3×Q8—182no—
173#237—182no—
174#237—184no—
175D3×D4—182no—
176D3×D4—184no—
177#235—91no—
178#235—91no—
179#235—91no—
180#235—182no—
181#235—181no—
182#235—184no—
183#235—181no—
184#235—362no—
185#235#15711yescharacteristic, complemented
186C2×D12—91no—
187C2×C4×D3—62no—
188C2×C4×D3—181no—
189C2×C4×D3—184no—
190C2×Dic6—91no—
191C3×M16—92no—
192C3×M16—181no—
193C2×C3×C8—181no—
194C4×Dic3—92no—
195C4×Dic3—181no—
196C22×C32—22no—
197C22×C32—42no—
198C22×C32—61no—
199C22×C32(C2×D3)⋊C411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—64no—
203C2×C3⋊D3—124no—
204C2×C3⋊D3—122no—
205C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
206C2×C3⋊D3C2×D1211yescharacteristic
207C2×C3×D3—24no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—124no—
211D32—128no—
212#157—64no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—42no—
216C32×C4—42no—
217C32×C4—61no—
218C32×C4(C2×D3)⋊C411yescharacteristic
219C32×C4C2×C4×D311yescharacteristic
220C3⋊Dic3—61no—
221C3⋊Dic3—64no—
222C3⋊Dic3—124no—
223C3⋊Dic3(C2×D3)⋊C411yescharacteristic
224C3⋊Dic3C2×#6711yescharacteristic
225C3×Dic3—24no—
226C3×Dic3—62no—
227C3×Dic3—62no—
228C2×#41—271no—
229C2×M16—91no—
230C2×C42—271no—
231#104—272no—
232#104—272no—
233C33C2×#10411yescharacteristic, complemented, Sylow
234C23×C3—91no—
235C22×D3—62no—
236C22×D3—181no—
237C22×D3—184no—
238C3×Q8—62no—
239C3×D4—62no—
240C3×D4—64no—
241C2×C3×C4—22no—
242C2×C3×C4—42no—
243C2×C3×C4—64no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—91no—
249C2×C3×C4C2×#15711yescharacteristic
250#67—21no—
251#67—181no—
252#67—184no—
253#67—181no—
254#67—184no—
255#67—184no—
256#67—181no—
257#67—364no—
258C2×Dic3—62no—
259C2×Dic3—181no—
260C2×Dic3—184no—
261C2×Dic3—182no—
262C2×Dic3—184no—
263D12—91no—
264D12—182no—
265D12—184no—
266D12—181no—
267D12—362no—
268D12C2×#15711yescharacteristic, complemented
269C4×D3—21no—
270C4×D3—62no—
271C4×D3—62no—
272C4×D3—62no—
273C4×D3—62no—
274C4×D3—181no—
275C4×D3—184no—
276C4×D3—184no—
277C4×D3—184no—
278C4×D3—181no—
279C4×D3—181no—
280C4×D3—364no—
281Dic6—91no—
282Dic6—182no—
283Dic6—181no—
284Dic6—184no—
285Dic6—362no—
286Dic6C2×#15711yescharacteristic, complemented
287C3×C8—182no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—42no—
291C2×C32—42no—
292C2×C32—82no—
293C2×C32—121no—
294C2×C32C2×(C2×D3)⋊C411yescharacteristic
295C3⋊D3—21no—
296C3⋊D3—121no—
297C3⋊D3—124no—
298C3⋊D3—244no—
299C3⋊D3#82512yescomplemented
300C3×D3—44no—
301C3×D3—64no—
302C3×D3—122no—
303#41—31no—
304#41—271no—
305#41—271no—
306#41—271no—
307#41—541no—
308#41—541no—
309C2×Q8—271no—
310C2×D4—271no—
311C2×D4—541no—
312C22×C4—91no—
313C22×C4—541no—
314C22×C4—541no—
315M16—92no—
316M16—181no—
317C2×C8—181no—
318C42—272no—
319C42—541no—
320C22×C3—22no—
321C22×C3—42no—
322C22×C3—64no—
323C22×C3—91no—
324C22×C3—91no—
325C22×C3—182no—
326C22×C3C22⋊#15711yescharacteristic
327C2×D3—21no—
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—124no—
331C2×D3—184no—
332C2×D3—181no—
333C2×D3—364no—
334C2×D3—362no—
335C2×D3—368no—
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4—64no—
341C3×C4—91no—
342C3×C4—91no—
343C3×C4C22⋊#15711yescharacteristic
344C3×C4C22×#15711yescharacteristic
345Dic3—21no—
346Dic3—62no—
347Dic3—62no—
348Dic3—184no—
349Dic3—184no—
350Dic3—181no—
351Dic3—364no—
352C32—22no—
353C32—42no—
354C32C2×#82511yescharacteristic, complemented
355C23—91no—
356C23—541no—
357Q8—31no—
358Q8—271no—
359Q8—541no—
360D4—31no—
361D4—61no—
362D4—271no—
363D4—541no—
364D4—541no—
365D4—541no—
366D4—541no—
367C2×C4—61no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—541no—
374C2×C4—541no—
375C2×C4—541no—
376C2×C4—542no—
377C2×C4—544no—
378C2×C4—541no—
379C2×C4D3×#15711yescharacteristic
380C8—182no—
381C2×C3—21no—
382C2×C3—22no—
383C2×C3—42no—
384C2×C3—42no—
385C2×C3—82no—
386C2×C3—92no—
387C2×C3—124no—
388C2×C3—181no—
389C2×C3C2×C22⋊#15711yescharacteristic
390D3—41no—
391D3—64no—
392D3—122no—
393D3—364no—
394D3—361no—
395D3—724no—
396C22—61no—
397C22—91no—
398C22—91no—
399C22—182no—
400C22—541no—
401C22—1082no—
402C22(C2×D3)⋊#15711yescharacteristic
403C4—61no—
404C4—91no—
405C4—91no—
406C4—544no—
407C4—541no—
408C4(C2×D3)⋊#15711yescharacteristic
409C4C2×D3×#15711yescharacteristic
410C3—22no—
411C3—42no—
412C3#1059383911yescharacteristic, complemented
413C2—21no—
414C2—92no—
415C2—121no—
416C2—181no—
417C2—1081no—
418C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC4(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC3⋊D3×#235C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#157C2semidirect
3. C3×#64283C2semidirect
4. #11722712C2semidirect
5. #11722710C2semidirect
8. #89076C4semidirect
10. C32×#235C4semidirect
13. C3⋊D3×D12C4semidirect
17. C3⋊D3×Dic6C4semidirect
20. C4×C3⋊#157C22semidirect
22. C3×#3788C22semidirect
27. #64283D3semidirect
36. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
38. #6308C2×C4semidirect
45. C32×D12C2×C4semidirect
47. C32×Dic6C2×C4semidirect
84. #3788C2×D3semidirect
165. C3×C3⋊D3#104semidirect
185. #235#157semidirect
233. C33C2×#104semidirect
268. D12C2×#157semidirect
286. Dic6C2×#157semidirect
299. C3⋊D3#825semidirect
354. C32C2×#825semidirect
412. C3#10593839semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2352
C2×C4×C3⋊#1572
C3×#642832
#117227122
#117227102
#105938393
C2×#8259