Finite Group Databasebeta

← #13139195

Subgroups — #13139195

Order 1728.

607 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#11722491C211yesmaximal, characteristic
2M16×#272C211yesmaximal, characteristic, complemented
3#11721097C211yesmaximal, characteristic, complemented
4#63652⋊C3C212yesmaximal, complemented
5#11720322C212yesmaximal, complemented
6#10591005—32nomaximal
7C2×C4×#272C2211yescharacteristic
8#3828⋊C3C2212yes—
9#88677—21no—
10#88672—21no—
11#88655C411yescharacteristic, complemented
12C4×((C3⋊D3)⋊C3)⋊C2—21no—
13#88651C411yescharacteristic, complemented
14#88650—21no—
15#88649C412yescomplemented
16M16×H(3,𝔽3)C2211yescharacteristic, complemented
17#88538C2211yescharacteristic, complemented
18#88527—21no—
19C8×#272—21no—
20C4×(C3⋊Dic3)⋊C3C2212yescomplemented
21#88451—21no—
22#88449—21no—
23D3×#235—32no—
24C3×D3×M16—32no—
25C3×D3×M16—61no—
26#63855—32no—
27#63652D312yescomplemented
28C3×#825—32no—
29#63570—32no—
30#63568—32no—
31C22×#272D411yescharacteristic
32C2×((C3⋊D3)⋊C3)⋊C2—21no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084D411yescharacteristic
35C4×#272C2×C411yescharacteristic
36C4×#272C2×C411yescharacteristic
37C4×#272D411yescharacteristic
38C4×#272D411yescharacteristic
39#638⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C4⋊((C3⋊D3)⋊C3)C2×C412yes—
42C4×(C3⋊D3)⋊C3—22no—
43#634⋊C3C2×C412yes—
44#6200—22no—
45#6199—21no—
46#6198—21no—
47((C3⋊D3)⋊C3)⋊C4—22no—
48#6196—21no—
49C8×H(3,𝔽3)—21no—
50#6180—21no—
51#5139—91no—
52#4821—92no—
53#3828C2×D312yes—
54C3×#235—32no—
55C2×C3×C4×D3—32no—
56C2×C3×C4×D3—61no—
57D3×#67—62no—
58#3804—62no—
59D3×D12—32no—
60C4×D32—62no—
61#3799—32no—
62#3798—62no—
63#3795—32no—
64D3×Dic6—32no—
65C32×M16—21no—
66C32×M16C2×D312yescomplemented
67C4×C3⋊Dic3C2×D312yescomplemented
68C3×C4×Dic3—32no—
69C3×#208—32no—
70C3×#208—61no—
71C3×ΓL(1,𝔽25)—62no—
72C3×ΓL(1,𝔽25)—62no—
73C3×D3×C8—62no—
74C3×D3×C8—62no—
75Dic3⋊Dic3—62no—
76(C2×D3)⋊Dic3—62no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×#272—41no—
79C2×#272C22⋊C411yescharacteristic
80C2×#272C22⋊C411yescharacteristic
81C2×(C3⋊D3)⋊C3—22no—
82((C3⋊D3)⋊C3)⋊C2—42no—
83C4×H(3,𝔽3)C22×C411yescharacteristic
84C4×H(3,𝔽3)C2×D411yescharacteristic
85#1084C22⋊C411yescharacteristic
86#1084C22⋊C411yescharacteristic
87(C3⋊Dic3)⋊C3—22no—
88(C3⋊Dic3)⋊C3—42no—
89D3×#41—92no—
90C2×#235—91no—
91C2×C3×M16—91no—
92#926—91no—
93D3×M16—32no—
94D3×M16—61no—
95#872—92no—
96C3×#104—92no—
97#837—92no—
98#825—32no—
99#825—92no—
100#825—92no—
101#805—92no—
102C22×C3×D3—32no—
103C22×C3×D3—61no—
104C2×D32—62no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D312yes—
107#638—22no—
108C2×C3⋊Dic3—22no—
109C4⋊(C3⋊D3)C4×D312yes—
110C4×C3⋊D3—22no—
111#634C4×D312yes—
112C3×#67—62no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—62no—
116C3×D12—32no—
117C3×C4×D3—32no—
118C3×C4×D3—32no—
119C3×C4×D3—32no—
120C3×C4×D3—32no—
121C3×C4×D3—61no—
122C3×C4×D3—61no—
123C3×C4×D3—61no—
124C3×C4×D3—61no—
125C3×C4×D3—62no—
126C3×Dic6—32no—
127#627—62no—
128#626—62no—
129#626—62no—
130#625—62no—
131#624—62no—
132D3×Dic3—62no—
133D3×Dic3—62no—
134C32×C8—22no—
135C32×C8—22no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—62no—
138#421—271noSylow
139C2×H(3,𝔽3)—21no—
140C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
141#272—21no—
142#272—21no—
143(C3⋊D3)⋊C3—42no—
144C3×#41—92no—
145C22×C3×C4—91no—
146C2×#67—181no—
147#239—92no—
148D3×Q8—92no—
149#237—92no—
150#237—182no—
151D3×D4—92no—
152D3×D4—182no—
153#235—32no—
154#235—92no—
155#235—92no—
156#235—182no—
157#235—181no—
158#235—181no—
159C2×D12—91no—
160C2×C4×D3—32no—
161C2×C4×D3—61no—
162C2×C4×D3—182no—
163C2×C4×D3—181no—
164C2×Dic6—91no—
165C3×M16—32no—
166C3×M16—61no—
167C3×M16—91no—
168C3×M16—181no—
169C3×M16D3211yescharacteristic
170C2×C3×C8—181no—
171C3×C42—92no—
172#217—181no—
173(C2×D3)⋊C4—182no—
174C4⋊Dic3—181no—
175Dic3⋊C4—182no—
176C4×Dic3—32no—
177C4×Dic3—92no—
178#208—32no—
179#208—61no—
180ΓL(1,𝔽25)—62no—
181ΓL(1,𝔽25)—62no—
182D3×C8—62no—
183D3×C8—62no—
184C22×C32—21no—
185C22×C32D3×D412yes—
186C2×C3⋊D3—22no—
187C2×C3×D3—32no—
188C2×C3×D3—32no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C2×C3×D3—62no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194D32—124no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4C2×C4×D312yes—
198C32×C4D3×D412yes—
199C3⋊Dic3—22no—
200C3⋊Dic3—42no—
201C3×Dic3—32no—
202C3×Dic3—32no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—62no—
206C3×Dic3—122no—
207C2×#41—271no—
208C2×M16—91no—
209#117—271no—
210#104—92no—
211#104—272no—
212H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
213C23×C3—91no—
214C22×D3—32no—
215C22×D3—61no—
216C22×D3—182no—
217C22×D3—181no—
218C3×Q8—92no—
219C3×D4—92no—
220C3×D4—182no—
221C2×C3×C4—32no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4C2×D3211yescharacteristic
231#67—62no—
232#67—182no—
233#67—182no—
234#67—182no—
235#67—182no—
236C2×Dic3—32no—
237C2×Dic3—61no—
238C2×Dic3—62no—
239C2×Dic3—182no—
240C2×Dic3—182no—
241C2×Dic3—181no—
242D12—32no—
243D12—92no—
244D12—181no—
245D12—182no—
246C4×D3—32no—
247C4×D3—32no—
248C4×D3—32no—
249C4×D3—32no—
250C4×D3—61no—
251C4×D3—61no—
252C4×D3—61no—
253C4×D3—61no—
254C4×D3—62no—
255C4×D3—182no—
256C4×D3—182no—
257C4×D3—182no—
258C4×D3—182no—
259Dic6—32no—
260Dic6—92no—
261Dic6—182no—
262Dic6—181no—
263C3×C8—21no—
264C3×C8—62no—
265C3×C8—62no—
266C3×C8—181no—
267C3⋊C8—62no—
268C3⋊C8—62no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32—41no—
272C2×C32D3×C22⋊C412yes—
273C3⋊D3—42no—
274C3×D3—62no—
275C3×D3—62no—
276C3×D3—62no—
277C3×D3—121no—
278C3×D3—122no—
279#41—92no—
280#41—272no—
281#41—541no—
282#41—541no—
283C2×Q8—271no—
284C2×D4—271no—
285C2×D4—541no—
286C22×C4—91no—
287C22×C4—541no—
288M16—91no—
289M16—181no—
290M16((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
291C2×C8—181no—
292C4⋊C4—541no—
293C22⋊C4—541no—
294C42—92no—
295C22×C3—32no—
296C22×C3—61no—
297C22×C3—91no—
298C22×C3—91no—
299C22×C3—182no—
300C22×C3—361no—
301C22×C3#380211yescharacteristic
302C2×D3—32no—
303C2×D3—32no—
304C2×D3—61no—
305C2×D3—61no—
306C2×D3—62no—
307C2×D3—122no—
308C2×D3—122no—
309C2×D3—182no—
310C2×D3—364no—
311C2×D3—362no—
312C3×C4—32no—
313C3×C4—32no—
314C3×C4—61no—
315C3×C4—61no—
316C3×C4—91no—
317C3×C4—91no—
318C3×C4—182no—
319C3×C4—362no—
320C3×C4C4×D3211yescharacteristic
321C3×C4#381111yescharacteristic
322Dic3—32no—
323Dic3—32no—
324Dic3—61no—
325Dic3—61no—
326Dic3—62no—
327Dic3—122no—
328Dic3—182no—
329Dic3—362no—
330C32—21no—
331C32#482112yescomplemented
332C23—91no—
333C23—541no—
334Q8—92no—
335Q8—541no—
336D4—92no—
337D4—182no—
338D4—541no—
339D4—542no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4—91no—
345C2×C4—182no—
346C2×C4—182no—
347C2×C4—542no—
348C2×C4—541no—
349C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C8—21no—
351C8—181no—
352C2×C3—21no—
353C2×C3—32no—
354C2×C3—62no—
355C2×C3—61no—
356C2×C3—121no—
357C2×C3—181no—
358C2×C3—181no—
359C2×C3—362no—
360C2×C3#6392111yescharacteristic
361D3—62no—
362D3—62no—
363D3—62no—
364D3—121no—
365D3—122no—
366D3—362no—
367C22—91no—
368C22—91no—
369C22—182no—
370C22—361no—
371C22—1082no—
372C22#8865511yescharacteristic
373C4—91no—
374C4—91no—
375C4—182no—
376C4—362no—
377C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C4#8867711yescharacteristic
379C3—32no—
380C3—61no—
381C3#1059100911yescharacteristic
382C2—21no—
383C2—181no—
384C2—181no—
385C2—362no—
386C2#1172116111yescharacteristic
subgroupG/H
centerC4#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3#63921
prime radical#11722491C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#421
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×#272C2semidirect
3. #11721097C2semidirect
4. #63652⋊C3C2semidirect
5. #11720322C2semidirect
11. #88655C4semidirect
13. #88651C4semidirect
15. #88649C4semidirect
16. M16×H(3,𝔽3)C22semidirect
17. #88538C22semidirect
20. C4×(C3⋊Dic3)⋊C3C22semidirect
27. #63652D3semidirect
66. C32×M16C2×D3semidirect
67. C4×C3⋊Dic3C2×D3semidirect
212. H(3,𝔽3)#421semidirect
290. M16((C3⋊D3)⋊C3)⋊C2semidirect
331. C32#4821semidirect

Maximal subgroups

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

subgroupindex
#117224912
M16×#2722
#117210972
#63652⋊C32
#117203222
#105910053