Finite Group Databasebeta

← #13166363

Subgroups — #13166363

Order 1728.

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

normal subgroups only

This group has 732 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
2#11724511C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
4#11722708C212yesmaximal, complemented
5#11722706C212yesmaximal, complemented
6#10593840—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×D12C2211yescharacteristic
14C4×D3×C3⋊D3—21no—
15#89020—21no—
16#89018C411yescharacteristic, complemented
17C3⋊D3×Dic6C2211yescharacteristic
18#89014C411yescharacteristic, complemented
19#88993—21no—
20#88989—21no—
21#88988C2212yes—
22C3×C4⋊#157—21no—
23C3×C4⋊#157C2212yes—
24#41×C3⋊D3—31no—
25D3×#235—62no—
26C2×C4⋊#157D311yescharacteristic, complemented
27#64283—31no—
28#63784—32no—
29#63782—32no—
30C22×C3×C3⋊D3D411yescharacteristic
31C2×D3×C3⋊D3—21no—
32C2×C3×#157—22no—
33C2×C3×#157—42no—
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—
56#4966—91nomaximal
57C22×#157—21no—
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—61no—
82C4⋊#157—21no—
83C4⋊#157C2×D312yes—
84#3788—61no—
85#3787—32no—
86C22×C33C22⋊C411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3C22⋊C411yescharacteristic
91C2×C3×C3⋊D3C2×D411yescharacteristic
92C2×C32×D3—21no—
93D3×C3⋊D3—41no—
94D3×C3⋊D3—41no—
95C3×#157—42no—
96C33×C4C22⋊C411yescharacteristic
97C33×C4C22×C411yescharacteristic
98C3⋊(C3⋊Dic3)—21no—
99C3×C3⋊Dic3C22⋊C411yescharacteristic
100C3×C3⋊Dic3C2×D411yescharacteristic
101C32×Dic3—21no—
102D3×#41—182no—
103C2×#235—91no—
104C2×C3×C4⋊C4—91no—
105C2×#208—91no—
106#810—92no—
107#809—92no—
108C22×C3⋊D3—61no—
109C22×C3⋊D3#6711yescharacteristic
110C22×C3×D3—62no—
111C2×D32—64no—
112C2×#157—22no—
113C2×#157—42no—
114C32×Q8—31no—
115C32×D4—31no—
116C32×D4—61no—
117C2×C32×C4—22no—
118C2×C32×C4—42no—
119C2×C32×C4—61no—
120C2×C32×C4C4×D311yescharacteristic
121#638—61no—
122#638—61no—
123#638—61no—
124#638—64no—
125#638—124no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3D1211yescharacteristic
128C4⋊(C3⋊D3)—31no—
129C4⋊(C3⋊D3)—62no—
130C4⋊(C3⋊D3)—61no—
131C4⋊(C3⋊D3)—122no—
132C4×C3⋊D3—61no—
133C4×C3⋊D3—61no—
134C4×C3⋊D3—61no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—124no—
137C4×C3⋊D3C4×D311yescharacteristic
138C4×C3⋊D3D1211yescharacteristic
139C4×C3⋊D3#6711yescharacteristic
140C4×C3⋊D3C22×D311yescharacteristic
141#634—31no—
142#634—62no—
143#634—61no—
144#634—122no—
145C3×#67—24no—
146C2×C3×Dic3—62no—
147C3×D12—22no—
148C3×C4×D3—24no—
149C3×C4×D3—62no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×Dic6—22no—
154#627—64no—
155#626—64no—
156#626—64no—
157#625—64no—
158#624—64no—
159D3×Dic3—64no—
160D3×Dic3—64no—
161#622—62no—
162#417—271noSylow
163C2×C33—21no—
164C2×C33C2×C22⋊C411yescharacteristic
165C3⋊(C3⋊D3)—41no—
166C3×C3⋊D3—21no—
167C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
168C3×C3⋊D3#13711yescharacteristic
169C32×D3—41no—
170C3×#41—62no—
171C22×C3×C4—91no—
172C22×C3×C4—181no—
173C2×#67—181no—
174#239—182no—
175D3×Q8—182no—
176#237—182no—
177#237—184no—
178D3×D4—182no—
179D3×D4—184no—
180#235—91no—
181#235—91no—
182#235—91no—
183#235—182no—
184#235—181no—
185#235—184no—
186#235—181no—
187#235—362no—
188#235#15711yescharacteristic, complemented
189C2×D12—91no—
190C2×C4×D3—62no—
191C2×C4×D3—181no—
192C2×C4×D3—184no—
193C2×Dic6—91no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4—181no—
196#208—181no—
197#208—181no—
198C2×C3⋊C8—92no—
199C22×C32—22no—
200C22×C32—42no—
201C22×C32—61no—
202C22×C32(C2×D3)⋊C411yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3—61no—
206C2×C3⋊D3—64no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—121no—
209C2×C3⋊D3—121no—
210C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
211C2×C3⋊D3C2×#6711yescharacteristic
212C2×C3×D3—24no—
213C2×C3×D3—62no—
214C2×C3×D3—62no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217D32—124no—
218D32—124no—
219#157—42no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—42no—
223C32×C4—42no—
224C32×C4—61no—
225C32×C4(C2×D3)⋊C411yescharacteristic
226C32×C4C2×C4×D311yescharacteristic
227C3⋊Dic3—61no—
228C3⋊Dic3—64no—
229C3⋊Dic3—124no—
230C3⋊Dic3(C2×D3)⋊C411yescharacteristic
231C3⋊Dic3C2×D1211yescharacteristic
232C3×Dic3—24no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C2×#41—271no—
236C2×M16—271no—
237C2×C4⋊C4—91no—
238#103—272no—
239#102—272no—
240C33#41711yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C22×D3—62no—
243C22×D3—181no—
244C22×D3—184no—
245C3×Q8—62no—
246C3×D4—62no—
247C3×D4—64no—
248C2×C3×C4—22no—
249C2×C3×C4—42no—
250C2×C3×C4—64no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—182no—
257C2×C3×C4—362no—
258C2×C3×C4C2×#15711yescharacteristic
259#67—21no—
260#67—181no—
261#67—184no—
262#67—181no—
263#67—184no—
264#67—184no—
265#67—181no—
266#67—364no—
267C2×Dic3—62no—
268C2×Dic3—181no—
269C2×Dic3—184no—
270D12—91no—
271D12—182no—
272D12—184no—
273D12—181no—
274D12—362no—
275D12C2×#15711yescharacteristic, complemented
276C4×D3—21no—
277C4×D3—62no—
278C4×D3—62no—
279C4×D3—62no—
280C4×D3—62no—
281C4×D3—181no—
282C4×D3—184no—
283C4×D3—184no—
284C4×D3—184no—
285C4×D3—181no—
286C4×D3—181no—
287C4×D3—364no—
288Dic6—91no—
289Dic6—182no—
290Dic6—181no—
291Dic6—184no—
292Dic6—362no—
293Dic6C2×#15711yescharacteristic, complemented
294C3⋊C8—182no—
295C2×C32—22no—
296C2×C32—21no—
297C2×C32—42no—
298C2×C32—42no—
299C2×C32—82no—
300C2×C32—121no—
301C2×C32C2×(C2×D3)⋊C411yescharacteristic
302C3⋊D3—21no—
303C3⋊D3—121no—
304C3⋊D3—124no—
305C3⋊D3—244no—
306C3⋊D3#93211yescharacteristic
307C3⋊D3#94211yescharacteristic
308C3×D3—44no—
309C3×D3—62no—
310C3×D3—62no—
311C3×D3—122no—
312#41—31no—
313#41—271no—
314#41—271no—
315#41—271no—
316#41—541no—
317#41—541no—
318C2×Q8—271no—
319C2×D4—271no—
320C2×D4—541no—
321C22×C4—91no—
322C22×C4—181no—
323C22×C4—541no—
324M16—541no—
325M16—541no—
326C2×C8—272no—
327C4⋊C4—92no—
328C4⋊C4—181no—
329C22×C3—22no—
330C22×C3—42no—
331C22×C3—64no—
332C22×C3—91no—
333C22×C3—91no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3C22⋊#15711yescharacteristic
337C2×D3—21no—
338C2×D3—62no—
339C2×D3—62no—
340C2×D3—122no—
341C2×D3—122no—
342C2×D3—184no—
343C2×D3—181no—
344C2×D3—364no—
345C2×D3—361no—
346C2×D3—361no—
347C2×D3—364no—
348C2×D3—364no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—64no—
354C3×C4—91no—
355C3×C4—91no—
356C3×C4—362no—
357C3×C4C22⋊#15711yescharacteristic
358C3×C4C22×#15711yescharacteristic
359Dic3—21no—
360Dic3—62no—
361Dic3—62no—
362Dic3—184no—
363Dic3—181no—
364Dic3—364no—
365C32—22no—
366C32—42no—
367C32#496611yescharacteristic, complemented
368C23—91no—
369C23—541no—
370Q8—31no—
371Q8—271no—
372Q8—541no—
373D4—31no—
374D4—61no—
375D4—271no—
376D4—541no—
377D4—541no—
378D4—541no—
379D4—541no—
380C2×C4—61no—
381C2×C4—91no—
382C2×C4—91no—
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—91no—
386C2×C4—182no—
387C2×C4—362no—
388C2×C4—541no—
389C2×C4—541no—
390C2×C4—541no—
391C2×C4—541no—
392C2×C4D3×#15711yescharacteristic
393C8—542no—
394C2×C3—21no—
395C2×C3—22no—
396C2×C3—42no—
397C2×C3—42no—
398C2×C3—82no—
399C2×C3—91no—
400C2×C3—91no—
401C2×C3—124no—
402C2×C3—181no—
403C2×C3C2×C22⋊#15711yescharacteristic
404D3—41no—
405D3—62no—
406D3—62no—
407D3—122no—
408D3—364no—
409D3—361no—
410D3—724no—
411C22—61no—
412C22—91no—
413C22—91no—
414C22—181no—
415C22—181no—
416C22—541no—
417C22—1081no—
418C22—1081no—
419C22(C2×D3)⋊#15711yescharacteristic
420C4—61no—
421C4—91no—
422C4—91no—
423C4—362no—
424C4—541no—
425C4(C2×D3)⋊#15711yescharacteristic
426C4C2×D3×#15711yescharacteristic
427C3—22no—
428C3—42no—
429C3#1059384011yescharacteristic, complemented
430C2—21no—
431C2—91no—
432C2—91no—
433C2—121no—
434C2—181no—
435C2—1081no—
436C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(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
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724511C2semidirect
3. C2×C3×C4⋊#157C2semidirect
4. #11722708C2semidirect
5. #11722706C2semidirect
8. #89076C4semidirect
10. C32×#235C4semidirect
16. #89018C4semidirect
18. #89014C4semidirect
26. C2×C4⋊#157D3semidirect
36. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
38. #6308C2×C4semidirect
45. C32×D12C2×C4semidirect
47. C32×Dic6C2×C4semidirect
188. #235#157semidirect
240. C33#417semidirect
275. D12C2×#157semidirect
293. Dic6C2×#157semidirect
367. C32#4966semidirect
429. C3#10593840semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2352
#117245112
C2×C3×C4⋊#1572
#117227082
#117227062
#105938403
#49669