Finite Group Databasebeta

← #13162040

Subgroups — #13162040

Order 1728.

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

normal subgroups only

This group has 670 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×#64158C212yesmaximal, complemented
2C3×#64157C211yesmaximal, characteristic, complemented
3C2×#88810C211yesmaximal, characteristic, complemented
4#11722081C211yesmaximal, characteristic, complemented
5#11722073C212yesmaximal, complemented
6#10591678—31nomaximal
7#10591677—32nomaximal
8C33×#41C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C3×#3755—22no—
11C3×#3753—22no—
12C3×#3748—21no—
13C3×#3748C2211yescharacteristic
14C3×#3748C2212yes—
15C2×C3×C3⋊(C3⋊C8)—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17#88810—21no—
18#88810—21no—
19#64158D312yescomplemented
20#64157D311yescharacteristic, complemented
21C3×#949—32no—
22C3×#949—32no—
23C3×#949—62no—
24C3×#948—32no—
25C3×#948—61no—
26C2×#3714—32no—
27C2×#3713—31no—
28#63578—31no—
29#63577—32no—
30#63559—32no—
31#63558—32no—
32#63558—32no—
33C22×C3×C3⋊D3—22no—
34C33×Q8D411yescharacteristic
35C33×D4—21no—
36C33×D4D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C3×C4⋊(C3⋊D3)—22no—
40C3×C4⋊(C3⋊D3)—22no—
41C3×C3⋊(C3⋊C8)—21no—
42C3×C3⋊(C3⋊C8)—22no—
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C3×C3⋊(C3⋊C8)D411yescharacteristic
45#5199—92no—
46#5142—91no—
47C32×#41—21no—
48C32×#41—22no—
49C32×#41—41no—
50C32×#41C2×D311yescharacteristic
51C32×#41C2×D312yes—
52C2×C4⋊(C3⋊D3)C2×D312yes—
53C2×C3×D12—32no—
54C2×C3×D12—32no—
55C2×C3×D12—62no—
56#3755—22no—
57#3753—22no—
58#3748—21no—
59#3748C2×D311yescharacteristic
60#3748C2×D312yes—
61C2×C3⋊(C3⋊C8)—21no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#215—64no—
64C3×#215—62no—
65C3×#215—62no—
66C3×#213—64no—
67C3×#213—62no—
68C3×#213—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#3714—62no—
81#3714—62no—
82#3713—61no—
83#3713—61no—
84C22×C33—21no—
85C22×C33#4111yescharacteristic
86C2×C3×C3⋊D3—42no—
87C2×C3×C3⋊D3—82no—
88C33×C4—21no—
89C33×C4C2×D411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C3×AGL(1,ℤ/8)—92no—
92C3×#131—91no—
93#949—32no—
94#949—32no—
95#949—62no—
96#948—32no—
97#948—61no—
98C2×#213—92no—
99C2×D24—91no—
100#833—92no—
101#823—92no—
102#823—92no—
103#822—92no—
104#819—91no—
105C22×C3⋊D3—22no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C22×C3×D3—122no—
109C32×Q8—21no—
110C32×Q8—22no—
111C32×Q8—41no—
112C32×Q8D1211yescharacteristic
113C32×Q8#6712yes—
114C32×D4—21no—
115C32×D4—22no—
116C32×D4—22no—
117C32×D4—21no—
118C32×D4—22no—
119C32×D4—41no—
120C32×D4—42no—
121C32×D4—42no—
122C32×D4D1211yescharacteristic
123C32×D4#6712yes—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—21no—
128C2×C32×C4—22no—
129C2×C32×C4—41no—
130C2×C32×C4—42no—
131C2×C32×C4—42no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D312yes—
134C4⋊(C3⋊D3)—22no—
135C4⋊(C3⋊D3)—22no—
136C3×D12—64no—
137C3×D12—62no—
138C3×D12—62no—
139C3×D12—62no—
140C3×D12—62no—
141C3×D12—122no—
142C3⋊(C3⋊C8)—21no—
143C3⋊(C3⋊C8)—22no—
144C3⋊(C3⋊C8)#6711yescharacteristic
145C3⋊(C3⋊C8)#6711yescharacteristic
146C3×C3⋊C8—32no—
147C3×C3⋊C8—32no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—62no—
151C3×C3⋊C8—64no—
152C3×C3⋊C8—62no—
153C3×C3⋊C8—62no—
154C3×C3⋊C8—62no—
155#472—271noSylow
156C2×C33—21no—
157C2×C33—41no—
158C2×C33#12111yescharacteristic
159C3×C3⋊D3—82no—
160C3×#41—21no—
161C3×#41—22no—
162C3×#41—41no—
163C3×#41D3211yescharacteristic
164C3×#41D3212yes—
165C2×C3×D4—92no—
166C2×D12—32no—
167C2×D12—32no—
168C2×D12—62no—
169C3×SD16—182no—
170C3×D8—182no—
171C3×M16—92no—
172C3×M16—91no—
173C3×M16—181no—
174C2×C3×C8—91no—
175C2×C3×C8—181no—
176#215—64no—
177#215—62no—
178#215—62no—
179#213—64no—
180#213—62no—
181#213—62no—
182#213—182no—
183#213—182no—
184#208—32no—
185#208—32no—
186#208—32no—
187#208—61no—
188#208—62no—
189#208—62no—
190#208—62no—
191C2×C3⋊C8—32no—
192C2×C3⋊C8—61no—
193C2×C3⋊C8—62no—
194C2×C3⋊C8—62no—
195D24—181no—
196D24—181no—
197C22×C32—21no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32#23511yescharacteristic
206C22×C32#23712yes—
207C2×C3⋊D3—42no—
208C2×C3⋊D3—82no—
209C2×C3×D3—124no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C2×C3×D3—244no—
213C2×C3×D3—242no—
214C2×C3×D3—242no—
215C32×C4—21no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4—21no—
219C32×C4—22no—
220C32×C4—22no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—42no—
225C32×C4—42no—
226C32×C4C2×D1211yescharacteristic
227C32×C4D3×D412yes—
228C32×C4C2×#6712yes—
229C32×C4C2×#6711yescharacteristic
230AGL(1,ℤ/8)—92no—
231C2×D8—271no—
232#131—91no—
233#108—271no—
234#100—272no—
235C33#47211yescharacteristic, complemented, Sylow
236C23×C3—182no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—122no—
240C3×Q8—21no—
241C3×Q8—22no—
242C3×Q8—41no—
243C3×Q8#62511yescharacteristic
244C3×Q8#62612yes—
245C3×D4—22no—
246C3×D4—22no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—22no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—41no—
253C3×D4—182no—
254C3×D4—182no—
255C3×D4#62511yescharacteristic
256C3×D4#62612yes—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4C2×D3211yescharacteristic
266C2×C3×C4C2×D3212yes—
267D12—64no—
268D12—62no—
269D12—62no—
270D12—62no—
271D12—62no—
272D12—122no—
273C3×C8—91no—
274C3×C8—91no—
275C3×C8—182no—
276C3×C8—181no—
277C3⋊C8—32no—
278C3⋊C8—32no—
279C3⋊C8—61no—
280C3⋊C8—61no—
281C3⋊C8—62no—
282C3⋊C8—64no—
283C3⋊C8—62no—
284C3⋊C8—62no—
285C3⋊C8—62no—
286C2×C32—21no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—41no—
291C2×C32—42no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—42no—
295C2×C32—42no—
296C2×C32—81no—
297C2×C32—82no—
298C2×C32—82no—
299C2×C32#93011yescharacteristic
300C2×C32#93912yes—
301C3⋊D3—82no—
302C3×D3—244no—
303C3×D3—242no—
304C3×D3—242no—
305#41#111311yescharacteristic
306C2×D4—92no—
307SD16—182no—
308D8—182no—
309D8—541no—
310D8—541no—
311M16—91no—
312M16—92no—
313M16—181no—
314C2×C8—91no—
315C2×C8—181no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—362no—
325C22×C3—722no—
326C22×C3#380412yes—
327C22×C3#380511yescharacteristic
328C2×D3—124no—
329C2×D3—122no—
330C2×D3—122no—
331C2×D3—244no—
332C2×D3—242no—
333C2×D3—242no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—21no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—22no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—41no—
344C3×C4—41no—
345C3×C4C2×#62511yescharacteristic
346C3×C4C2×#62612yes—
347C3×C4D3×#6712yes—
348C3×C4#381111yescharacteristic
349C32—22no—
350C32—21no—
351C32—41no—
352C32#514211yescharacteristic, complemented
353C32#519912yescomplemented
354C23—182no—
355Q8#629511yescharacteristic
356D4—21no—
357D4—182no—
358D4—182no—
359D4#629511yescharacteristic
360C2×C4—21no—
361C2×C4C2×#111311yescharacteristic
362C8—91no—
363C8—91no—
364C8—182no—
365C8—181no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—22no—
370C2×C3—42no—
371C2×C3—41no—
372C2×C3—42no—
373C2×C3—42no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—81no—
377C2×C3—82no—
378C2×C3—82no—
379C2×C3—722no—
380C2×C3#6397812yes—
381C2×C3#6398011yescharacteristic
382D3—244no—
383D3—242no—
384D3—242no—
385C22—21no—
386C22—362no—
387C22—722no—
388C22#8904611yescharacteristic
389C4—21no—
390C4C2×#629511yescharacteristic
391C4#8904911yescharacteristic
392C3—22no—
393C3—21no—
394C3—41no—
395C3#1059167712yescomplemented
396C3#1059167811yescharacteristic, complemented
397C2—21no—
398C2—41no—
399C2—722no—
400C2#1172371711yescharacteristic
subgroupG/H
centerC2#11723717
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×#41C22
socleC2×C33#121
hypercenter#41#1113
hypocenterC33#472
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
1. C3×#64158C2semidirect
2. C3×#64157C2semidirect
3. C2×#88810C2semidirect
4. #11722081C2semidirect
5. #11722073C2semidirect
19. #64158D3semidirect
20. #64157D3semidirect
235. C33#472semidirect
352. C32#5142semidirect
353. C32#5199semidirect
395. C3#10591677semidirect
396. C3#10591678semidirect

Maximal subgroups

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

subgroupindex
C3×#641582
C3×#641572
C2×#888102
#117220812
#117220732
#105916783
#105916773