Finite Group Databasebeta

← #13162042

Subgroups — #13162042

Order 1728.

622 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 99 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×#64160C212yesmaximal
2C3×#64157C211yesmaximal, characteristic
3C2×#88812C211yesmaximal, characteristic, complemented
4#11722081C211yesmaximal, characteristic, complemented
5#11722074C212yesmaximal, complemented
6#10591683—31nomaximal
7#10591682—32nomaximal
8C33×#41C2211yescharacteristic
9C2×C3×#634C2212yes—
10C3×#3756—22no—
11C3×#3754—22no—
12C3×#3748—21no—
13C3×#3748C2211yescharacteristic
14C3×#3748C2212yes—
15C2×C3×C3⋊(C3⋊C8)—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17#88812—21no—
18#88812—21no—
19#64160D312yes—
20#64157D311yescharacteristic
21C3×#951—32no—
22C3×#951—32no—
23C3×#951—62no—
24C3×#948—32no—
25C3×#948—61no—
26C2×#3719—32no—
27C2×#3718—31no—
28#63578—31no—
29#63577—32no—
30#63561—32no—
31#63560—32no—
32#63560—32no—
33C33×Q8D411yescharacteristic
34C33×D4—21no—
35C33×D4D411yescharacteristic
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3×C3⋊Dic3—22no—
39C3×#634—22no—
40C3×#634—22no—
41C3×C3⋊(C3⋊C8)—21no—
42C3×C3⋊(C3⋊C8)—22no—
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C3×C3⋊(C3⋊C8)D411yescharacteristic
45#5204—92no—
46#5143—91no—
47C32×#41—21no—
48C32×#41—22no—
49C32×#41—41no—
50C32×#41C2×D311yescharacteristic
51C32×#41C2×D312yes—
52C2×#634C2×D312yes—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—62no—
56#3756—22no—
57#3754—22no—
58#3748—21no—
59#3748C2×D311yescharacteristic
60#3748C2×D312yes—
61C2×C3⋊(C3⋊C8)—21no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#216—64no—
64C3×#216—62no—
65C3×#216—62no—
66C3×#214—64no—
67C3×#214—62no—
68C3×#214—62no—
69C3×#208—32no—
70C3×#208—32no—
71C3×#208—32no—
72C3×#208—62no—
73C3×#208—61no—
74C3×#208—62no—
75C3×#208—62no—
76C2×C3×C3⋊C8—32no—
77C2×C3×C3⋊C8—62no—
78C2×C3×C3⋊C8—61no—
79C2×C3×C3⋊C8—62no—
80#3719—62no—
81#3719—62no—
82#3718—61no—
83#3718—61no—
84C22×C33—21no—
85C22×C33#4111yescharacteristic
86C33×C4—21no—
87C33×C4C2×D411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3—42no—
90C3×#137—92no—
91C3×#131—91no—
92#951—32no—
93#951—32no—
94#951—62no—
95#948—32no—
96#948—61no—
97C2×#216—92no—
98C2×Dic12—91no—
99#836—92no—
100#824—92no—
101#824—92no—
102#822—92no—
103#819—91no—
104C32×Q8—21no—
105C32×Q8—22no—
106C32×Q8—41no—
107C32×Q8D1211yescharacteristic
108C32×Q8#6712yes—
109C32×D4—21no—
110C32×D4—22no—
111C32×D4—22no—
112C32×D4—21no—
113C32×D4—22no—
114C32×D4—41no—
115C32×D4—42no—
116C32×D4—42no—
117C32×D4D1211yescharacteristic
118C32×D4#6712yes—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4—41no—
125C2×C32×C4—42no—
126C2×C32×C4—42no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D312yes—
129C2×C3⋊Dic3—22no—
130#634—22no—
131#634—22no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—122no—
135C3×Dic6—64no—
136C3×Dic6—62no—
137C3×Dic6—62no—
138C3×Dic6—62no—
139C3×Dic6—62no—
140C3×Dic6—122no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—22no—
143C3⋊(C3⋊C8)#6711yescharacteristic
144C3⋊(C3⋊C8)#6711yescharacteristic
145C3×C3⋊C8—32no—
146C3×C3⋊C8—32no—
147C3×C3⋊C8—61no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—64no—
151C3×C3⋊C8—62no—
152C3×C3⋊C8—62no—
153C3×C3⋊C8—62no—
154#473—271noSylow
155C2×C33—21no—
156C2×C33—41no—
157C2×C33#12111yescharacteristic
158C3×#41—21no—
159C3×#41—22no—
160C3×#41—41no—
161C3×#41D3211yescharacteristic
162C3×#41D3212yes—
163C2×C3×Q8—92no—
164C2×Dic6—32no—
165C2×Dic6—32no—
166C2×Dic6—62no—
167C3×Q16—182no—
168C3×SD16—182no—
169C3×M16—92no—
170C3×M16—91no—
171C3×M16—181no—
172C2×C3×C8—91no—
173C2×C3×C8—181no—
174#216—64no—
175#216—62no—
176#216—62no—
177#216—182no—
178#216—182no—
179#214—64no—
180#214—62no—
181#214—62no—
182#208—32no—
183#208—32no—
184#208—32no—
185#208—61no—
186#208—62no—
187#208—62no—
188#208—62no—
189C2×C3⋊C8—32no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—62no—
192C2×C3⋊C8—62no—
193Dic12—181no—
194Dic12—181no—
195C22×C32—21no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—42no—
202C22×C32—42no—
203C22×C32#23511yescharacteristic
204C22×C32#23712yes—
205C32×C4—21no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—21no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4—21no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4—42no—
215C32×C4—42no—
216C32×C4C2×D1211yescharacteristic
217C32×C4D3×D412yes—
218C32×C4C2×#6712yes—
219C32×C4C2×#6711yescharacteristic
220C3⋊Dic3—42no—
221C3×Dic3—124no—
222C3×Dic3—122no—
223C3×Dic3—122no—
224#137—92no—
225C2×Q16—271no—
226#131—91no—
227#108—271no—
228#101—272no—
229C33#47311yescharacteristic, complemented, Sylow
230C3×Q8—21no—
231C3×Q8—22no—
232C3×Q8—41no—
233C3×Q8—182no—
234C3×Q8—182no—
235C3×Q8#62511yescharacteristic
236C3×Q8#62612yes—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—22no—
242C3×D4—42no—
243C3×D4—42no—
244C3×D4—41no—
245C3×D4#62511yescharacteristic
246C3×D4#62612yes—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—182no—
256C2×C3×C4C2×D3211yescharacteristic
257C2×C3×C4C2×D3212yes—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—122no—
261Dic6—64no—
262Dic6—62no—
263Dic6—62no—
264Dic6—62no—
265Dic6—62no—
266Dic6—122no—
267C3×C8—91no—
268C3×C8—91no—
269C3×C8—182no—
270C3×C8—181no—
271C3⋊C8—32no—
272C3⋊C8—32no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—62no—
276C3⋊C8—64no—
277C3⋊C8—62no—
278C3⋊C8—62no—
279C3⋊C8—62no—
280C2×C32—21no—
281C2×C32—22no—
282C2×C32—22no—
283C2×C32—21no—
284C2×C32—41no—
285C2×C32—42no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—42no—
289C2×C32—42no—
290C2×C32—81no—
291C2×C32—82no—
292C2×C32—82no—
293C2×C32#93011yescharacteristic
294C2×C32#93912yes—
295#41#111311yescharacteristic
296C2×Q8—92no—
297Q16—182no—
298Q16—541no—
299Q16—541no—
300SD16—182no—
301M16—91no—
302M16—92no—
303M16—181no—
304C2×C8—91no—
305C2×C8—181no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3#380412yes—
315C22×C3#380511yescharacteristic
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—21no—
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—22no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4—41no—
326C3×C4—41no—
327C3×C4—362no—
328C3×C4C2×#62511yescharacteristic
329C3×C4C2×#62612yes—
330C3×C4D3×#6712yes—
331C3×C4#381111yescharacteristic
332Dic3—124no—
333Dic3—122no—
334Dic3—122no—
335C32—22no—
336C32—21no—
337C32—41no—
338C32#514311yescharacteristic, complemented
339C32#520412yescomplemented
340Q8—182no—
341Q8—182no—
342Q8#629511yescharacteristic
343D4—21no—
344D4#629511yescharacteristic
345C2×C4—21no—
346C2×C4—182no—
347C2×C4C2×#111311yescharacteristic
348C8—91no—
349C8—91no—
350C8—182no—
351C8—181no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—22no—
355C2×C3—21no—
356C2×C3—41no—
357C2×C3—42no—
358C2×C3—42no—
359C2×C3—42no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—81no—
363C2×C3—82no—
364C2×C3—82no—
365C2×C3#6397812yes—
366C2×C3#6398011yescharacteristic
367C22—21no—
368C22#8904611yescharacteristic
369C4—21no—
370C4—362no—
371C4C2×#629511yescharacteristic
372C4#8904911yescharacteristic
373C3—21no—
374C3—22no—
375C3—41no—
376C3#1059168212yescomplemented
377C3#1059168311yescharacteristic, complemented
378C2—21no—
379C2—41no—
380C2#1172371711yescharacteristic
subgroupG/H
centerC2#11723717
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
Fitting, prime radicalC33×#41C22
socleC2×C33#121
hypercenter#41#1113
hypocenterC33#473
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
3. C2×#88812C2semidirect
4. #11722081C2semidirect
5. #11722074C2semidirect
229. C33#473semidirect
338. C32#5143semidirect
339. C32#5204semidirect
376. C3#10591682semidirect
377. C3#10591683semidirect

Maximal subgroups

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

subgroupindex
C3×#641602
C3×#641572
C2×#888122
#117220812
#117220742
#105916833
#105916823