Finite Group Databasebeta

← M16×#627 #11861528

Subgroups — M16×#627 #11861528

Order 1152.

847 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#627C211yesmaximal, characteristic
2M16×C3⋊Dic3C211yesmaximal, characteristic
3C3×Dic3×M16C212yesmaximal
4#10591120C211yesmaximal, characteristic
5#10591115C212yesmaximal
6#10590787C212yesmaximal, complemented
7#10590784C214yesmaximal, complemented
8C8×#627C212yesmaximal, complemented
9M16×Dic6—32nomaximal
10C22×#627C411yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C4×C3⋊Dic3C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×C3×C4×Dic3C2212yes—
15C2×C3×#208C2212yes—
16C2×#3724C411yescharacteristic
17C2×Dic3⋊Dic3C412yes—
18C4×#627C411yescharacteristic
19C4×#627C412yes—
20C4×#627C411yescharacteristic
21C4×#627C2211yescharacteristic
22C4×#627C2212yes—
23C4×#627C2211yescharacteristic
24#63642C2212yes—
25C8×C3⋊Dic3C2212yes—
26C3×#815C2214yes—
27C3×C8×Dic3C2214yes—
28#63572C2212yes—
29#63572C2212yes—
30Dic3⋊(C3⋊C8)C2214yes—
31Dic3⋊(C3⋊C8)C2214yes—
32C2×C4×Dic6—32no—
33C3×C4×M16—32no—
34#5118—32no—
35#5118—32no—
36Dic3×M16—32no—
37Dic3×M16—61no—
38Dic3×M16D312yes—
39#4924—32no—
40#4701—34no—
41#4701—34no—
42#4679—34no—
43C8×Dic6—34no—
44C22×C32×C4C2311yescharacteristic
45C22×C3⋊Dic3C2×C411yescharacteristic
46C22×C3⋊Dic3C2×C411yescharacteristic
47C22×C3×Dic3C2×C412yes—
48C22×C3×Dic3C2×C412yes—
49C2×#627—24no—
50C2×#627C2×C411yescharacteristic
51C2×#627C2×C412yes—
52C2×#627C2×C411yescharacteristic
53C32×M16Q814yescomplemented
54C2×C32×C8C2312yes—
55C4×C3⋊Dic3C2×C411yescharacteristic
56C4×C3⋊Dic3C2×C411yescharacteristic
57C4×C3⋊Dic3C2311yescharacteristic
58C4×C3⋊Dic3C2311yescharacteristic
59#3748—22no—
60C2×C3⋊(C3⋊C8)C2312yes—
61C3×C4×Dic3C2×C412yes—
62C3×C4×Dic3C2×C412yes—
63C3×C4×Dic3C2312yes—
64C3×C4×Dic3C2312yes—
65C3×#208—24no—
66C2×C3×C3⋊C8C2314yes—
67#3724C2×C412yes—
68#3724C2×C411yescharacteristic
69#3724C2×C411yescharacteristic
70Dic3⋊Dic3C2×C412yes—
71Dic3⋊Dic3C2×C412yes—
72Dic3⋊Dic3C2×C412yes—
73Dic3⋊Dic3C2×C412yes—
74Q8×M16—91noSylow
75C22×Dic6—32no—
76C2×C3×M16—21no—
77C2×C3×M16—32no—
78C2×C3×M16C2×D312yes—
79C2×C3×C42—32no—
80C2×C4⋊Dic3—32no—
81C2×Dic3⋊C4—32no—
82C2×Dic3⋊C4—32no—
83C2×C4×Dic3—32no—
84C2×C4×Dic3—61no—
85C2×C4×Dic3C2×D312yes—
86C2×#208—32no—
87C2×#208—61no—
88C2×#208C2×D312yes—
89C4×Dic6—32no—
90C4×Dic6—32no—
91C4×Dic6—32no—
92C4×Dic6—32no—
93C4×Dic6—32no—
94C4×Dic6—32no—
95C4×Dic6—32no—
96C4×Dic6—32no—
97C3×#97—34no—
98C3×C4×C8—34no—
99#815—34no—
100#815—62no—
101#815C2×D314yes—
102Dic3⋊C8—34no—
103Dic3⋊C8—34no—
104Dic3⋊C8—34no—
105Dic3⋊C8—34no—
106C8×Dic3—34no—
107C8×Dic3—62no—
108C8×Dic3C2×D314yes—
109C4⋊(C3⋊C8)—34no—
110C4⋊(C3⋊C8)—34no—
111C23×C32C22×C411yescharacteristic
112C2×C32×C4C22×C411yescharacteristic
113C2×C32×C4C2×Q812yes—
114C2×C32×C4#4112yes—
115C2×C32×C4C2411yescharacteristic
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—41no—
118C2×C3⋊Dic3C22×C411yescharacteristic
119C2×C3⋊Dic3C22×C411yescharacteristic
120C2×C3⋊Dic3C22×C411yescharacteristic
121C2×C3⋊Dic3C22×C411yescharacteristic
122C2×C3×Dic3—24no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3C22×C412yes—
125C2×C3×Dic3C22×C412yes—
126C2×C3×Dic3C22×C412yes—
127C2×C3×Dic3C22×C412yes—
128#627—22no—
129#627—24no—
130#627M1614yesdirect product factor
131C32×C8C2×Q814yescomplemented
132C3⋊(C3⋊C8)—22no—
133C3×C3⋊C8—24no—
134C2×C4×Q8—91no—
135#446—94no—
136#446—92no—
137C8×Q8—92no—
138#423—92no—
139#423—91no—
140C4×M16—32no—
141C4×M16—91no—
142C22×C3×C4—21no—
143C22×C3×C4—32no—
144C22×C3×C4—32no—
145C22×C3×C4C22×D312yes—
146C22×Dic3—32no—
147C22×Dic3—32no—
148C22×Dic3—61no—
149C22×Dic3—61no—
150C22×Dic3C4×D312yes—
151C22×Dic3C4×D312yes—
152C2×Dic6—32no—
153C2×Dic6—32no—
154C2×Dic6—32no—
155C2×Dic6—32no—
156C2×Dic6—68no—
157C3×M16—24no—
158C3×M16—64no—
159C3×M16Dic618yescomplemented
160C2×C3×C8—22no—
161C2×C3×C8—34no—
162C2×C3×C8C22×D314yes—
163C3×C42—32no—
164C3×C42—32no—
165C3×C42—32no—
166C3×C42—32no—
167C4⋊Dic3—32no—
168C4⋊Dic3—32no—
169C4⋊Dic3—32no—
170C4⋊Dic3—32no—
171Dic3⋊C4—32no—
172Dic3⋊C4—32no—
173Dic3⋊C4—32no—
174Dic3⋊C4—32no—
175Dic3⋊C4—32no—
176Dic3⋊C4—32no—
177Dic3⋊C4—32no—
178Dic3⋊C4—32no—
179C4×Dic3—32no—
180C4×Dic3—32no—
181C4×Dic3—32no—
182C4×Dic3—32no—
183C4×Dic3—61no—
184C4×Dic3—61no—
185C4×Dic3—61no—
186C4×Dic3—61no—
187C4×Dic3C4×D312yes—
188C4×Dic3C4×D312yes—
189C4×Dic3C22×D312yes—
190C4×Dic3C22×D312yes—
191#208—24no—
192#208—64no—
193#208—64no—
194C2×C3⋊C8—34no—
195C2×C3⋊C8—62no—
196C2×C3⋊C8C22×D314yes—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32C4×Q812yes—
200C22×C32C23×C411yescharacteristic
201C32×C4C4×Q812yes—
202C32×C4C22×Q811yescharacteristic
203C32×C4C2×#4111yescharacteristic
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3C2×M1612yes—
208C3×Dic3—22no—
209C3×Dic3—22no—
210C3×Dic3—22no—
211C3×Dic3C2×M1614yes—
212C22×Q8—91no—
213C2×M16—32no—
214C2×M16—91no—
215C2×M16D3211yescharacteristic
216C4×Q8—91no—
217C4×Q8—92no—
218C4×Q8—91no—
219C4×Q8—91no—
220C4×Q8—92no—
221C4×Q8—91no—
222C2×C4⋊C4—91no—
223C2×C4⋊C4—92no—
224C2×C42—32no—
225C2×C42—91no—
226C4⋊C8—94no—
227C4⋊C8—94no—
228C4⋊C8—92no—
229C4⋊C8—92no—
230#97—34no—
231#97—92no—
232C4×C8—34no—
233C4×C8—92no—
234C23×C3—21no—
235C23×C3C2×C4×D312yes—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—32no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—64no—
245C2×C3×C4—122no—
246C2×C3×C4C2×Dic614yes—
247C2×C3×C4C2×C4×D312yes—
248C2×C3×C4#23514yes—
249C2×C3×C4C23×D312yes—
250C2×Dic3—24no—
251C2×Dic3—32no—
252C2×Dic3—32no—
253C2×Dic3—32no—
254C2×Dic3—32no—
255C2×Dic3—42no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—64no—
261C2×Dic3—122no—
262C2×Dic3—122no—
263C2×Dic3—122no—
264C2×Dic3C2×C4×D312yes—
265C2×Dic3C2×C4×D312yes—
266C2×Dic3C2×C4×D312yes—
267C2×Dic3C2×C4×D312yes—
268Dic6—38no—
269Dic6—64no—
270Dic6—64no—
271Dic6—64no—
272C3×C8—24no—
273C3×C8—64no—
274C3×C8C2×Dic618yescomplemented
275C3⋊C8—24no—
276C3⋊C8—64no—
277C3⋊C8—64no—
278C2×C32—22no—
279C2×C32C2×C4×Q811yescharacteristic
280C2×C32C22×M1611yescharacteristic
281C2×C32#56811yescharacteristic
282C2×Q8—91no—
283C2×Q8—92no—
284C2×Q8—91no—
285C2×Q8—184no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—91no—
289C22×C4—91no—
290C22×C4C2×D3211yescharacteristic
291M16—64no—
292M16—182no—
293M16#62714yesdirect product factor
294C2×C8—34no—
295C2×C8—92no—
296C2×C8C2×D3212yes—
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4—92no—
300C4⋊C4—92no—
301C4⋊C4—92no—
302C4⋊C4—92no—
303C4⋊C4—92no—
304C42—32no—
305C42—32no—
306C42—32no—
307C42—32no—
308C42—91no—
309C42—91no—
310C42—91no—
311C42—91no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—21no—
315C22×C3—22no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3C4×Dic614yes—
319C22×C3C22×C4×D312yes—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—22no—
323C3×C4—34no—
324C3×C4—62no—
325C3×C4—62no—
326C3×C4—62no—
327C3×C4C4×Dic614yes—
328C3×C4C22×Dic612yes—
329C3×C4C2×#23512yes—
330Dic3—22no—
331Dic3—22no—
332Dic3—22no—
333Dic3—34no—
334Dic3—62no—
335Dic3—62no—
336Dic3—62no—
337Dic3—62no—
338Dic3—62no—
339Dic3—121no—
340Dic3—121no—
341Dic3D3×M1614yes—
342C32Q8×M1611yescharacteristic, complemented, Sylow
343C23C4×D3211yescharacteristic
344Q8—94no—
345Q8—184no—
346Q8—182no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—64no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—91no—
355C2×C4—91no—
356C2×C4—122no—
357C2×C4—182no—
358C2×C4—361no—
359C2×C4#379812yes—
360C2×C4C4×D3211yescharacteristic
361C2×C4C2×#62712yes—
362C2×C4C22×D3211yescharacteristic
363C8—64no—
364C8—182no—
365C8C2×#62714yescomplemented
366C2×C3—24no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—42no—
371C2×C3C2×C4×Dic612yes—
372C2×C3C2×D3×M1612yes—
373C2×C3#574512yes—
374C22—21no—
375C22—21no—
376C22C4×#62712yes—
377C22C2×C4×D3211yescharacteristic
378C4—34no—
379C4—62no—
380C4—62no—
381C4—62no—
382C4—92no—
383C4—181no—
384C4—181no—
385C4—181no—
386C4C4×#62712yes—
387C4C2×#379811yescharacteristic
388C4C22×#62711yescharacteristic
389C3—21no—
390C3M16×Dic612yescomplemented
391C2—22no—
392C2D32×M1611yescharacteristic
393C2#1059472211yescharacteristic
394C2C2×C4×#62711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC2×C32×M16C22
prime radicalC23×C32C22×C4
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32Q8×M16
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
6. #10590787C2semidirect
7. #10590784C2semidirect
8. C8×#627C2semidirect
53. C32×M16Q8semidirect
130. #627M16direct (N × H)
131. C32×C8C2×Q8semidirect
159. C3×M16Dic6semidirect
274. C3×C8C2×Dic6semidirect
293. M16#627direct (N × H)
342. C32Q8×M16semidirect
365. C8C2×#627semidirect
390. C3M16×Dic6semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6272
M16×C3⋊Dic32
C3×Dic3×M162
#105911202
#105911152
#105907872
#105907842
C8×#6272
M16×Dic63