Finite Group Databasebeta

← #16855077

Subgroups — #16855077

Order 3968.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#11758567C211yesmaximal, characteristic, complemented
2#14521849C211yesmaximal, characteristic, complemented
3#14521847C211yesmaximal, characteristic, complemented
4#14521625C211yesmaximal, characteristic, complemented
5#14521233C214yesmaximal, complemented
6#14521223C214yesmaximal, complemented
7#14521067C211yesmaximal, characteristic, complemented
8C2×#11758412C211yesmaximal, characteristic, complemented
9C31×#421C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11#11758567—22no—
12#11758567—22no—
13#11758567—22no—
14#11758567C2214yes—
15C2×Q8×D31—21no—
16C2×#92497—21no—
17C2×#92497—22no—
18C2×#92495—21no—
19C2×#92495C2211yescharacteristic
20C2×#92495C2211yescharacteristic
21C22×Dic62—21no—
22C22×Dic62C2211yescharacteristic
23C2×M16×C31C2211yescharacteristic
24C31×#117C2211yescharacteristic
25#11758528C2212yescomplemented
26#11758527—22no—
27#11758526C2212yescomplemented
28#11758519—21no—
29#11758509—21no—
30C2×#92474C2211yescharacteristic
31#11758486—21no—
32#11758486C2212yescomplemented
33#11758485—21no—
34#11758485C2212yescomplemented
35#11758481—22no—
36#11758474—22no—
37#11758463—22no—
38#11758452C2212yescomplemented
39#11758446C2212yescomplemented
40C31×#104C2214yescomplemented
41#11758412C2214yescomplemented
42#11758393C2214yescomplemented
43#41×C31—21no—
44#41×C31—21no—
45#41×C31D412yes—
46#41×C31C2312yes—
47C2×Q8×C31D411yescharacteristic
48C2×D4×C31—21no—
49C2×D4×C31D411yescharacteristic
50C22×C4×C31—21no—
51C22×C4×C31C2311yescharacteristic
52C2×C22⋊D31—22no—
53C2×C22⋊D31—21no—
54C2×C22⋊D31—21no—
55C22×Dic31—22no—
56C22×Dic31D412yes—
57Q8×D31—24no—
58Q8×D31—24no—
59#92497—28no—
60#92497—24no—
61#92497—24no—
62#92497—28no—
63#92495—24no—
64#92495—21no—
65#92495—21no—
66#92495—24no—
67#92495—21no—
68#92495—21no—
69#92495—21no—
70#92495—21no—
71#92495D412yes—
72#92495C2312yes—
73C2×D124D411yescharacteristic
74C2×D124C2311yescharacteristic
75C2×C4×D31—22no—
76C2×C4×D31—21no—
77C2×C4×D31—21no—
78C2×Dic62—22no—
79C2×Dic62—24no—
80C2×Dic62—21no—
81C2×Dic62—21no—
82C2×Dic62—21no—
83C2×Dic62—21no—
84C2×Dic62—24no—
85C2×Dic62D412yes—
86C2×Dic62D411yescharacteristic
87C2×Dic62C2311yescharacteristic
88C2×Dic62C2312yes—
89M16×C31—21no—
90M16×C31C2312yes—
91C2×C8×C31—21no—
92C4⋊C4×C31—21no—
93C22⋊C4×C31—21no—
94C42×C31C2312yes—
95#92482—22no—
96#92481—22no—
97#92480—22no—
98#92479—22no—
99(C2×D31)⋊C4—24no—
100C4⋊Dic31—24no—
101#92474—21no—
102#92474C2312yes—
103C2×C31⋊C8—21no—
104Dic124—22no—
105D248—22no—
106#92470—22no—
107#92470—22no—
108C23×C31—21no—
109C23×C31C2×D411yescharacteristic
110C22×D31—21no—
111C22×D31—21no—
112Q8×C31—21no—
113Q8×C31C2×D412yes—
114D4×C31—22no—
115D4×C31—22no—
116D4×C31—21no—
117D4×C31C2×D412yes—
118C2×C4×C31—21no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—22no—
122C2×C4×C31C2×D411yescharacteristic
123C2×C4×C31C2×D411yescharacteristic
124C2×C4×C31C2×D411yescharacteristic
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31C2×D411yescharacteristic
127C2×C4×C31C2411yescharacteristic
128C22⋊D31—28no—
129C22⋊D31—22no—
130C22⋊D31—22no—
131C22⋊D31—41no—
132C22⋊D31—41no—
133C2×Dic31—28no—
134C2×Dic31—22no—
135C2×Dic31—21no—
136C2×Dic31—22no—
137C2×Dic31—21no—
138C2×Dic31C2×D414yes—
139D124—21no—
140D124—21no—
141D124—21no—
142D124C2×D412yes—
143C4×D31—28no—
144C4×D31—22no—
145C4×D31—22no—
146C4×D31—41no—
147C4×D31—41no—
148Dic62—28no—
149Dic62—22no—
150Dic62—21no—
151Dic62—22no—
152Dic62—22no—
153Dic62—22no—
154Dic62—21no—
155Dic62—21no—
156Dic62C2×D412yes—
157C8×C31—22no—
158C31⋊C8—22no—
159#3020—311nomaximal, Sylow
160C22×C31—22no—
161C22×C31—41no—
162C22×C31—42no—
163C22×C31#12011yescharacteristic
164C22×C31#12011yescharacteristic
165C22×C31C22×D411yescharacteristic
166C2×D31—22no—
167C2×D31—42no—
168C2×D31—42no—
169C2×D31—41no—
170C4×C31—22no—
171C4×C31—42no—
172C4×C31#12011yescharacteristic
173C4×C31#12011yescharacteristic
174C4×C31C22×D411yescharacteristic
175C4×C31C22×D411yescharacteristic
176Dic31—24no—
177Dic31—22no—
178Dic31—41no—
179C2×2−1+4—311no—
180#575—311no—
181#575—311no—
182#536—311no—
183#456—314no—
184#454—314no—
185#421—311no—
186#421D3111yescharacteristic, complemented
187C2×#101—311no—
188C2×C31—21no—
189C2×C31—21no—
190C2×C31—21no—
191C2×C31—42no—
192C2×C31C2×#12011yescharacteristic
193D31—42no—
194D31—81no—
1952−1+4—314no—
1962−1+4—622no—
1972−1+4—622no—
1982−1+4—622no—
199C2×#41—311no—
200C2×#41—311no—
201C2×#41—621no—
202C2×#41—622no—
203C2×#41—621no—
204C2×#41C2×D3111yescharacteristic
205C22×Q8—311no—
206C22×Q8—621no—
207C22×Q8—621no—
208#137—312no—
209#137—312no—
210#137—621no—
211#137—621no—
212AGL(1,ℤ/8)—312no—
213AGL(1,ℤ/8)—312no—
214AGL(1,ℤ/8)—621no—
215AGL(1,ℤ/8)—621no—
216#135—622no—
217#135—622no—
218C2×M16—311no—
219C2×M16C2×D3111yescharacteristic
220C4⋊Q8—312no—
221#124—312no—
222#123—622no—
223C22⋊Q8—622no—
224#117C2×D3111yescharacteristic
225#104—314no—
226#104C2×D3114yescomplemented
227#101—314no—
228C31#302011yescharacteristic, complemented, Sylow
229#41—21no—
230#41—21no—
231#41—312no—
232#41—312no—
233#41—624no—
234#41—624no—
235#41—628no—
236#41—628no—
237#41—624no—
238#41—624no—
239#41—621no—
240#41—621no—
241#41—621no—
242#41—621no—
243#41—621no—
244#41—621no—
245#41D12412yes—
246#41C22×D3112yes—
247C2×Q8—312no—
248C2×Q8—312no—
249C2×Q8—311no—
250C2×Q8—311no—
251C2×Q8—624no—
252C2×Q8—622no—
253C2×Q8—624no—
254C2×Q8—624no—
255C2×Q8—624no—
256C2×Q8—621no—
257C2×Q8—621no—
258C2×Q8—621no—
259C2×Q8—621no—
260C2×Q8D12411yescharacteristic
261C2×D4—21no—
262C2×D4—311no—
263C2×D4—311no—
264C2×D4—622no—
265C2×D4—621no—
266C2×D4—621no—
267C2×D4D12411yescharacteristic
268C22×C4—21no—
269C22×C4—312no—
270C22×C4—622no—
271C22×C4—622no—
272C22×C4—621no—
273C22×C4—621no—
274C22×C4C22×D3111yescharacteristic
275Q16—622no—
276Q16—622no—
277SD16—622no—
278SD16—622no—
279SD16—622no—
280SD16—622no—
281D8—622no—
282D8—622no—
283M16—21no—
284M16—312no—
285M16—621no—
286M16C22×D3112yes—
287C2×C8—21no—
288C2×C8—621no—
289C4⋊C4—21no—
290C4⋊C4—624no—
291C22⋊C4—21no—
292C22⋊C4—624no—
293C42C22×D3112yes—
294C23—21no—
295C23—621no—
296C23—621no—
297C23D4×D3111yescharacteristic
298Q8—21no—
299Q8—312no—
300Q8—628no—
301Q8—622no—
302Q8—622no—
303Q8—621no—
304Q8—622no—
305Q8—622no—
306Q8—621no—
307Q8—621no—
308Q8C2×D12412yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—312no—
313D4—628no—
314D4—622no—
315D4—622no—
316D4—621no—
317D4—621no—
318D4—621no—
319D4—1241no—
320D4—1241no—
321D4C2×D12412yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—314no—
327C2×C4—628no—
328C2×C4—628no—
329C2×C4—622no—
330C2×C4—622no—
331C2×C4—621no—
332C2×C4—622no—
333C2×C4—622no—
334C2×C4—621no—
335C2×C4—1241no—
336C2×C4—1241no—
337C2×C4C2×D12411yescharacteristic
338C2×C4C2×D12411yescharacteristic
339C2×C4D4×D3111yescharacteristic
340C2×C4D4×D3111yescharacteristic
341C2×C4D4×D3111yescharacteristic
342C2×C4C23×D3111yescharacteristic
343C8—22no—
344C8—622no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—622no—
349C22—1242no—
350C22—1242no—
351C22—1241no—
352C22#1175845911yescharacteristic
353C22#1175845911yescharacteristic
354C22C2×D4×D3111yescharacteristic
355C4—22no—
356C4—42no—
357C4—624no—
358C4—622no—
359C4—1241no—
360C4#1175845911yescharacteristic
361C4#1175845911yescharacteristic
362C4C22×D12411yescharacteristic
363C4C2×D4×D3111yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—1242no—
369C2—2481no—
370C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#421C2
socleC2×C31C2×#120
hypercenter#421D31
hypocenterC31#3020
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#11758567C2semidirect
2. #14521849C2semidirect
3. #14521847C2semidirect
4. #14521625C2semidirect
5. #14521233C2semidirect
6. #14521223C2semidirect
7. #14521067C2semidirect
8. C2×#11758412C2semidirect
9. C31×#421C2semidirect
25. #11758528C22semidirect
27. #11758526C22semidirect
32. #11758486C22semidirect
34. #11758485C22semidirect
38. #11758452C22semidirect
39. #11758446C22semidirect
40. C31×#104C22semidirect
41. #11758412C22semidirect
42. #11758393C22semidirect
186. #421D31semidirect
226. #104C2×D31semidirect
228. C31#3020semidirect

Maximal subgroups

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

subgroupindex
C2×#117585672
#145218492
#145218472
#145216252
#145212332
#145212232
#145210672
C2×#117584122
C31×#4212
#302031