Finite Group Databasebeta

← #13173438

Subgroups — #13173438

Order 1728.

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

normal subgroups only

This group has 732 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⋊#157)C211yesmaximal, characteristic, complemented
3#11724511C211yesmaximal, characteristic, complemented
4#11723109C212yesmaximal, complemented
5#11723107C212yesmaximal, complemented
6#10593840—31nomaximal
7C22×C3⋊#157—21no—
8C33×#41C411yescharacteristic, complemented
9C3×#3832C411yescharacteristic, complemented
10C3×Q8×C3⋊D3C2211yescharacteristic
11C3×#3830—21no—
12C3×#3830C411yescharacteristic, complemented
13C3×D4×C3⋊D3—21no—
14C3×D4×C3⋊D3C2211yescharacteristic
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⋊#157)C2212yes—
22#88989—21no—
23#88988C2212yes—
24#41×C3⋊D3D311yescharacteristic
25C3×D3×#41—62no—
26C2×C4⋊#157—31no—
27#64283—31no—
28#63784—32no—
29#63782—32no—
30C22×C3×C3⋊D3—21no—
31C22×C3×C3⋊D3D411yescharacteristic
32C2×C3⋊#157—22no—
33C2×C3⋊#157—42no—
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—
56#5232—91nomaximal
57C22×#157—61no—
58C32×#41—22no—
59C32×#41—42no—
60C32×#41Dic311yescharacteristic, complemented
61#3832Dic311yescharacteristic, complemented
62Q8×C3⋊D3C2×D311yescharacteristic
63#3830—21no—
64#3830Dic311yescharacteristic, complemented
65D4×C3⋊D3—21no—
66D4×C3⋊D3C2×D311yescharacteristic
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—61no—
85#3787—32no—
86C22×C33—21no—
87C22×C33C22⋊C411yescharacteristic
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3C22⋊C411yescharacteristic
94C2×C3×C3⋊D3C2×D411yescharacteristic
95C3⋊#157—42no—
96C33×C4—21no—
97C33×C4C22⋊C411yescharacteristic
98C33×C4C22×C411yescharacteristic
99C3×C3⋊Dic3—21no—
100C3×C3⋊Dic3C22⋊C411yescharacteristic
101C3×C3⋊Dic3C2×D411yescharacteristic
102C2×C3×#41—91no—
103D3×#41—62no—
104C2×C4⋊Dic3—91no—
105C2×#208—91no—
106#835—92no—
107#832—92no—
108C22×C3⋊D3—21no—
109C22×C3⋊D3#6711yescharacteristic
110C22×C3×D3—62no—
111C22×C3×D3—64no—
112C2×#157—62no—
113C2×#157—122no—
114C32×Q8—22no—
115C32×Q8—42no—
116C32×Q8C2×Dic311yescharacteristic, complemented
117C32×D4—21no—
118C32×D4—24no—
119C32×D4—22no—
120C32×D4—44no—
121C32×D4—42no—
122C32×D4C2×Dic311yescharacteristic, complemented
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—22no—
126C2×C32×C4—44no—
127C2×C32×C4—42no—
128C2×C32×C4C2×Dic311yescharacteristic
129#638—21no—
130#638—21no—
131#638—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6711yescharacteristic
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)C2×Dic311yescharacteristic, complemented
136C4×C3⋊D3—21no—
137C4×C3⋊D3—21no—
138C4×C3⋊D3—21no—
139C4×C3⋊D3C2×Dic311yescharacteristic
140C4×C3⋊D3#6711yescharacteristic
141C4×C3⋊D3#6711yescharacteristic
142C4×C3⋊D3C22×D311yescharacteristic
143#634—21no—
144#634C2×Dic311yescharacteristic, complemented
145C3×#67—64no—
146C3×#67—64no—
147C3×#67—64no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—64no—
150C3×D12—62no—
151C3×D12—64no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×C4×D3—64no—
157C3×C4×D3—64no—
158C3×C4×D3—64no—
159C3×Dic6—62no—
160C3×Dic6—64no—
161#622—62no—
162#417—271noSylow
163C2×C33—21no—
164C2×C33—41no—
165C2×C33C2×C22⋊C411yescharacteristic
166C3×C3⋊D3—21no—
167C3×C3⋊D3—41no—
168C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
169C3×C3⋊D3#13711yescharacteristic
170C3×#41—22no—
171C3×#41—42no—
172C3×#41—91no—
173C3×#41—91no—
174C3×#41—91no—
175C3×#41—181no—
176C3×#41—181no—
177C3×#41#15711yescharacteristic, complemented
178C2×C3×Q8—91no—
179C2×C3×D4—91no—
180C2×C3×D4—181no—
181C22×C3×C4—91no—
182C22×C3×C4—181no—
183C22×Dic3—181no—
184#239—62no—
185D3×Q8—62no—
186#237—62no—
187#237—64no—
188D3×D4—62no—
189D3×D4—64no—
190#235—62no—
191#235—64no—
192C2×C4×D3—62no—
193C2×C4×D3—64no—
194C4⋊Dic3—92no—
195C4⋊Dic3—181no—
196#208—181no—
197#208—181no—
198C2×C3⋊C8—92no—
199C22×C32—21no—
200C22×C32—24no—
201C22×C32—22no—
202C22×C32—44no—
203C22×C32—42no—
204C22×C32#21711yescharacteristic
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3—21no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3⋊D3#21711yescharacteristic
211C2×C3⋊D3C2×#6711yescharacteristic
212C2×C3×D3—62no—
213C2×C3×D3—62no—
214C2×C3×D3—64no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C2×C3×D3—124no—
218C2×C3×D3—124no—
219#157—122no—
220C32×C4—21no—
221C32×C4—24no—
222C32×C4—22no—
223C32×C4—22no—
224C32×C4—44no—
225C32×C4—42no—
226C32×C4—42no—
227C32×C4#21711yescharacteristic
228C32×C4C22×Dic311yescharacteristic
229C3⋊Dic3—21no—
230C3⋊Dic3#21711yescharacteristic
231C3⋊Dic3C2×#6711yescharacteristic
232C3×Dic3—62no—
233C3×Dic3—62no—
234C3×Dic3—64no—
235C2×#41—91no—
236C2×M16—271no—
237C2×C4⋊C4—271no—
238#103—272no—
239#102—272no—
240C33#41711yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C23×C3—181no—
243C22×D3—62no—
244C22×D3—64no—
245C3×Q8—22no—
246C3×Q8—42no—
247C3×Q8—91no—
248C3×Q8—181no—
249C3×Q8C2×#15711yescharacteristic, complemented
250C3×D4—24no—
251C3×D4—21no—
252C3×D4—22no—
253C3×D4—44no—
254C3×D4—42no—
255C3×D4—91no—
256C3×D4—181no—
257C3×D4—181no—
258C3×D4—181no—
259C3×D4—181no—
260C3×D4C2×#15711yescharacteristic, complemented
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—22no—
264C2×C3×C4—44no—
265C2×C3×C4—42no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4C2×#15711yescharacteristic
276#67—64no—
277#67—64no—
278#67—64no—
279C2×Dic3—62no—
280C2×Dic3—64no—
281C2×Dic3—182no—
282C2×Dic3—362no—
283D12—62no—
284D12—64no—
285C4×D3—62no—
286C4×D3—62no—
287C4×D3—62no—
288C4×D3—62no—
289C4×D3—64no—
290C4×D3—64no—
291C4×D3—64no—
292Dic6—62no—
293Dic6—64no—
294C3⋊C8—182no—
295C2×C32—22no—
296C2×C32—21no—
297C2×C32—42no—
298C2×C32—42no—
299C2×C32—41no—
300C2×C32—44no—
301C2×C32—84no—
302C2×C32—82no—
303C2×C32C2×#21711yescharacteristic
304C3⋊D3—21no—
305C3⋊D3—41no—
306C3⋊D3#94911yescharacteristic
307C3⋊D3#95111yescharacteristic
308C3×D3—62no—
309C3×D3—62no—
310C3×D3—122no—
311C3×D3—124no—
312#41—91no—
313#41—91no—
314#41—91no—
315#41—181no—
316#41—181no—
317#41C3⋊#15711yescharacteristic, complemented
318C2×Q8—91no—
319C2×D4—91no—
320C2×D4—181no—
321C22×C4—91no—
322C22×C4—181no—
323C22×C4—541no—
324M16—541no—
325M16—541no—
326C2×C8—272no—
327C4⋊C4—272no—
328C4⋊C4—541no—
329C22×C3—24no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—44no—
333C22×C3—42no—
334C22×C3—91no—
335C22×C3—91no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3—181no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3C22⋊#15711yescharacteristic
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—64no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—124no—
348C2×D3—124no—
349C3×C4—24no—
350C3×C4—21no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—44no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—91no—
357C3×C4—91no—
358C3×C4—181no—
359C3×C4C22⋊#15711yescharacteristic
360C3×C4C22×#15711yescharacteristic
361Dic3—62no—
362Dic3—62no—
363Dic3—64no—
364Dic3—362no—
365C32—22no—
366C32—42no—
367C32#523211yescharacteristic, complemented
368C23—91no—
369C23—181no—
370Q8—91no—
371Q8—181no—
372Q8C2×C3⋊#15711yescharacteristic, complemented
373D4—21no—
374D4—91no—
375D4—181no—
376D4—181no—
377D4—181no—
378D4—181no—
379D4C2×C3⋊#15711yescharacteristic, complemented
380C2×C4—21no—
381C2×C4—91no—
382C2×C4—91no—
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—91no—
386C2×C4—181no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—542no—
391C2×C4—1082no—
392C2×C4C2×C3⋊#15711yescharacteristic
393C8—542no—
394C2×C3—21no—
395C2×C3—22no—
396C2×C3—42no—
397C2×C3—44no—
398C2×C3—41no—
399C2×C3—42no—
400C2×C3—82no—
401C2×C3—84no—
402C2×C3—91no—
403C2×C3—91no—
404C2×C3—181no—
405C2×C3—361no—
406C2×C3C2×C22⋊#15711yescharacteristic
407D3—62no—
408D3—62no—
409D3—122no—
410D3—124no—
411C22—21no—
412C22—91no—
413C22—91no—
414C22—181no—
415C22—181no—
416C22—181no—
417C22—361no—
418C22—361no—
419C22#8899411yescharacteristic
420C4—21no—
421C4—91no—
422C4—91no—
423C4—181no—
424C4—1082no—
425C4#8899411yescharacteristic
426C4C22×C3⋊#15711yescharacteristic
427C3—22no—
428C3—42no—
429C3#1059384011yescharacteristic, complemented
430C2—21no—
431C2—41no—
432C2—91no—
433C2—91no—
434C2—181no—
435C2—361no—
436C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#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
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊#157)C2semidirect
3. #11724511C2semidirect
4. #11723109C2semidirect
5. #11723107C2semidirect
8. C33×#41C4semidirect
9. C3×#3832C4semidirect
12. C3×#3830C4semidirect
16. C3×#3828C4semidirect
34. C33×Q8C2×C4semidirect
36. C33×D4C2×C4semidirect
45. C3×C4⋊(C3⋊D3)C2×C4semidirect
54. C3×#634C2×C4semidirect
60. C32×#41Dic3semidirect
61. #3832Dic3semidirect
64. #3830Dic3semidirect
68. #3828Dic3semidirect
116. C32×Q8C2×Dic3semidirect
122. C32×D4C2×Dic3semidirect
135. C4⋊(C3⋊D3)C2×Dic3semidirect
144. #634C2×Dic3semidirect
177. C3×#41#157semidirect
240. C33#417semidirect
249. C3×Q8C2×#157semidirect
260. C3×D4C2×#157semidirect
317. #41C3⋊#157semidirect
367. C32#5232semidirect
372. Q8C2×C3⋊#157semidirect
379. D4C2×C3⋊#157semidirect
429. C3#10593840semidirect

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⋊#157)2
#117245112
#117231092
#117231072
#105938403
#52329