Finite Group Databasebeta

← #13173439

Subgroups — #13173439

Order 1728.

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

normal subgroups only

This group has 724 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
2#11724514C211yesmaximal, characteristic, complemented
3C2×#88988C211yesmaximal, characteristic, complemented
4#11723109C212yesmaximal, complemented
5#11723107C212yesmaximal, complemented
6#10593841—31nomaximal
7C33×#41C411yescharacteristic, complemented
8C3×#3832C411yescharacteristic, complemented
9C3×Q8×C3⋊D3C2211yescharacteristic
10C3×#3830—21no—
11C3×#3830C411yescharacteristic, complemented
12C3×D4×C3⋊D3—21no—
13C3×D4×C3⋊D3C2211yescharacteristic
14C3×#3828—21no—
15C3×#3828C411yescharacteristic, complemented
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18#88994—21no—
19C2×C3⋊#622—21no—
20C4⋊(C3⋊#157)C2212yes—
21C4×C3⋊#157—21no—
22#88988—21no—
23#88988C2212yes—
24#41×C3⋊D3D311yescharacteristic
25C3×D3×#41—62no—
26#64286—31no—
27C2×#3787—31no—
28#63784—32no—
29#63782—32no—
30C22×C3×C3⋊D3—21no—
31C22×C3×C3⋊D3D411yescharacteristic
32C2×C3⋊#157—22no—
33C33×Q8C2×C411yescharacteristic
34C33×D4—21no—
35C33×D4C2×C411yescharacteristic
36C2×C33×C4—21no—
37C2×C33×C4C2×C411yescharacteristic
38C3×#638—21no—
39C3×#638—21no—
40C3×#638—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3C2×C411yescharacteristic
49C3×C4×C3⋊D3D411yescharacteristic
50C3×C4×C3⋊D3D411yescharacteristic
51C3×C4×C3⋊D3C2311yescharacteristic
52C3×#634—21no—
53C3×#634C2×C411yescharacteristic
54C3⋊#622—22no—
55#5233—91nomaximal
56C32×#41—22no—
57C32×#41—42no—
58C32×#41Dic311yescharacteristic, complemented
59#3832Dic311yescharacteristic, complemented
60Q8×C3⋊D3C2×D311yescharacteristic
61#3830—21no—
62#3830Dic311yescharacteristic, complemented
63D4×C3⋊D3—21no—
64D4×C3⋊D3C2×D311yescharacteristic
65#3828—21no—
66#3828Dic311yescharacteristic, complemented
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C3×#239—62no—
70C3×D3×Q8—62no—
71C3×#237—62no—
72C3×#237—64no—
73C3×D3×D4—62no—
74C3×D3×D4—64no—
75C3×#235—62no—
76C3×#235—64no—
77C2×C3×C4×D3—62no—
78C2×C3×C4×D3—64no—
79C22⋊#157—61no—
80C2×#622—61no—
81C4⋊#157—32no—
82C4×#157—61no—
83#3787—32no—
84#3787—61no—
85C22×C33—21no—
86C22×C33C22⋊C411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—42no—
90C2×C3×C3⋊D3C22⋊C411yescharacteristic
91C2×C3×C3⋊D3C2×D411yescharacteristic
92C3⋊#157—42no—
93C33×C4—21no—
94C33×C4C22⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3C22⋊C411yescharacteristic
98C3×C3⋊Dic3C2×D411yescharacteristic
99C2×C3×#41—91no—
100D3×#41—62no—
101#926—91no—
102C22×C3⋊C8—91no—
103#835—92no—
104#832—92no—
105C22×C3⋊D3—21no—
106C22×C3⋊D3#6711yescharacteristic
107C22×C3×D3—62no—
108C22×C3×D3—64no—
109C2×#157—62no—
110C32×Q8—22no—
111C32×Q8—42no—
112C32×Q8C2×Dic311yescharacteristic
113C32×D4—21no—
114C32×D4—24no—
115C32×D4—22no—
116C32×D4—44no—
117C32×D4—42no—
118C32×D4C2×Dic311yescharacteristic
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—22no—
122C2×C32×C4—44no—
123C2×C32×C4—42no—
124C2×C32×C4C2×Dic311yescharacteristic
125#638—21no—
126#638—21no—
127#638—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3#6711yescharacteristic
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)C2×Dic311yescharacteristic
132C4×C3⋊D3—21no—
133C4×C3⋊D3—21no—
134C4×C3⋊D3—21no—
135C4×C3⋊D3C2×Dic311yescharacteristic
136C4×C3⋊D3#6711yescharacteristic
137C4×C3⋊D3#6711yescharacteristic
138C4×C3⋊D3C22×D311yescharacteristic
139#634—21no—
140#634C2×Dic311yescharacteristic
141C3×#67—64no—
142C3×#67—64no—
143C3×#67—64no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—64no—
146C3×D12—62no—
147C3×D12—64no—
148C3×C4×D3—62no—
149C3×C4×D3—62no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×C4×D3—64no—
153C3×C4×D3—64no—
154C3×C4×D3—64no—
155C3×Dic6—62no—
156C3×Dic6—64no—
157#622—62no—
158#416—271noSylow
159C2×C33—21no—
160C2×C33—41no—
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—21no—
163C3×C3⋊D3—41no—
164C3×C3⋊D3#13512yes—
165C3×#41—22no—
166C3×#41—42no—
167C3×#41—91no—
168C3×#41—91no—
169C3×#41—91no—
170C3×#41—181no—
171C3×#41—181no—
172C3×#41#15711yescharacteristic, complemented
173C2×C3×Q8—91no—
174C2×C3×D4—91no—
175C2×C3×D4—181no—
176C22×C3×C4—91no—
177C22×C3×C4—181no—
178#239—62no—
179D3×Q8—62no—
180#237—62no—
181#237—64no—
182D3×D4—62no—
183D3×D4—64no—
184#235—62no—
185#235—64no—
186C2×C4×D3—62no—
187C2×C4×D3—64no—
188#217—181no—
189C4⋊Dic3—92no—
190C4×Dic3—181no—
191C2×C3⋊C8—92no—
192C2×C3⋊C8—181no—
193C2×C3⋊C8—181no—
194C22×C32—21no—
195C22×C32—24no—
196C22×C32—22no—
197C22×C32—44no—
198C22×C32—42no—
199C22×C32#21711yescharacteristic
200C2×C3⋊D3—21no—
201C2×C3⋊D3—22no—
202C2×C3⋊D3—42no—
203C2×C3⋊D3#21711yescharacteristic
204C2×C3⋊D3C2×#6711yescharacteristic
205C2×C3×D3—62no—
206C2×C3×D3—62no—
207C2×C3×D3—64no—
208C2×C3×D3—124no—
209C2×C3×D3—128no—
210#157—122no—
211C32×C4—21no—
212C32×C4—24no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—44no—
216C32×C4—42no—
217C32×C4—42no—
218C32×C4#21711yescharacteristic
219C32×C4C22×Dic311yescharacteristic
220C3⋊Dic3—21no—
221C3⋊Dic3#21711yescharacteristic
222C3⋊Dic3C2×#6711yescharacteristic
223C3×Dic3—62no—
224C3×Dic3—62no—
225C3×Dic3—64no—
226C2×#41—91no—
227C22×C8—271no—
228#117—271no—
229#103—272no—
230#102—272no—
231C33#41611yescharacteristic, complemented, Sylow
232C23×C3—91no—
233C23×C3—181no—
234C22×D3—62no—
235C22×D3—64no—
236C3×Q8—22no—
237C3×Q8—42no—
238C3×Q8—91no—
239C3×Q8—181no—
240C3×Q8C2×#15711yescharacteristic
241C3×D4—24no—
242C3×D4—21no—
243C3×D4—22no—
244C3×D4—44no—
245C3×D4—42no—
246C3×D4—91no—
247C3×D4—181no—
248C3×D4—181no—
249C3×D4—181no—
250C3×D4—181no—
251C3×D4C2×#15711yescharacteristic
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—44no—
256C2×C3×C4—42no—
257C2×C3×C4—91no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4C2×#15711yescharacteristic
267#67—64no—
268#67—64no—
269#67—64no—
270C2×Dic3—62no—
271C2×Dic3—64no—
272C2×Dic3—182no—
273D12—62no—
274D12—64no—
275C4×D3—62no—
276C4×D3—62no—
277C4×D3—62no—
278C4×D3—62no—
279C4×D3—64no—
280C4×D3—64no—
281C4×D3—64no—
282Dic6—62no—
283Dic6—64no—
284C3⋊C8—182no—
285C2×C32—22no—
286C2×C32—21no—
287C2×C32—42no—
288C2×C32—42no—
289C2×C32—41no—
290C2×C32—44no—
291C2×C32—84no—
292C2×C32—82no—
293C2×C32C2×#21711yescharacteristic
294C3⋊D3—21no—
295C3⋊D3—41no—
296C3⋊D3#95012yes—
297C3×D3—64no—
298C3×D3—122no—
299C3×D3—124no—
300#41—91no—
301#41—91no—
302#41—91no—
303#41—181no—
304#41—181no—
305#41C3⋊#15711yescharacteristic, complemented
306C2×Q8—91no—
307C2×D4—91no—
308C2×D4—181no—
309C22×C4—91no—
310C22×C4—181no—
311C2×C8—272no—
312C2×C8—541no—
313C2×C8—541no—
314C4⋊C4—272no—
315C22⋊C4—541no—
316C42—541no—
317C22×C3—24no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—44no—
321C22×C3—42no—
322C22×C3—91no—
323C22×C3—91no—
324C22×C3—182no—
325C22×C3—181no—
326C22×C3—362no—
327C22×C3C22⋊#15711yescharacteristic
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—64no—
331C2×D3—124no—
332C2×D3—128no—
333C3×C4—24no—
334C3×C4—21no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—44no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4—91no—
341C3×C4—91no—
342C3×C4—181no—
343C3×C4C22⋊#15711yescharacteristic
344C3×C4C22×#15711yescharacteristic
345Dic3—62no—
346Dic3—62no—
347Dic3—64no—
348Dic3—362no—
349C32—22no—
350C32—42no—
351C32#523311yescharacteristic, complemented
352C23—91no—
353C23—181no—
354Q8—91no—
355Q8—181no—
356Q8C2×C3⋊#15711yescharacteristic
357D4—21no—
358D4—91no—
359D4—181no—
360D4—181no—
361D4—181no—
362D4—181no—
363D4C2×C3⋊#15711yescharacteristic
364C2×C4—21no—
365C2×C4—91no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4—542no—
375C2×C4C2×C3⋊#15711yescharacteristic
376C8—542no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—42no—
380C2×C3—44no—
381C2×C3—41no—
382C2×C3—42no—
383C2×C3—82no—
384C2×C3—84no—
385C2×C3—92no—
386C2×C3—181no—
387C2×C3—361no—
388C2×C3C2×C22⋊#15711yescharacteristic
389D3—64no—
390D3—122no—
391D3—124no—
392C22—21no—
393C22—91no—
394C22—91no—
395C22—182no—
396C22—181no—
397C22—362no—
398C22#8899411yescharacteristic
399C4—21no—
400C4—91no—
401C4—91no—
402C4—181no—
403C4—1082no—
404C4#8899411yescharacteristic
405C4C22×C3⋊#15711yescharacteristic
406C3—22no—
407C3—42no—
408C3#1059384111yescharacteristic, complemented
409C2—21no—
410C2—41no—
411C2—92no—
412C2—181no—
413C2—361no—
414C2C2×#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
hypocenterC33#416
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. #11724514C2semidirect
3. C2×#88988C2semidirect
4. #11723109C2semidirect
5. #11723107C2semidirect
7. C33×#41C4semidirect
8. C3×#3832C4semidirect
11. C3×#3830C4semidirect
15. C3×#3828C4semidirect
58. C32×#41Dic3semidirect
59. #3832Dic3semidirect
62. #3830Dic3semidirect
66. #3828Dic3semidirect
172. C3×#41#157semidirect
231. C33#416semidirect
305. #41C3⋊#157semidirect
351. C32#5233semidirect
408. C3#10593841semidirect

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
#117245142
C2×#889882
#117231092
#117231072
#105938413
#52339