Finite Group Databasebeta

← #16855073

Subgroups — #16855073

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#14521847C211yesmaximal, characteristic, complemented
3#14521620C211yesmaximal, characteristic, complemented
4#14521230C214yesmaximal, complemented
5#14521220C214yesmaximal, complemented
6#14521066C211yesmaximal, characteristic, complemented
7#14521065C211yesmaximal, characteristic, complemented
8C2×#11758412C211yesmaximal, characteristic, complemented
9C31×#575C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C2×#41×C31C2211yescharacteristic
12#11758567—22no—
13#11758567—22no—
14#11758567—22no—
15#11758567C2214yes—
16C2×Q8×D31—21no—
17C2×#92497—21no—
18C2×#92497—22no—
19C2×#92495—21no—
20C2×#92495C2211yescharacteristic
21C22×Dic62—21no—
22C22×Dic62C2211yescharacteristic
23C31×#137—21no—
24C31×#137C2212yescomplemented
25C31×AGL(1,ℤ/8)—21no—
26C31×AGL(1,ℤ/8)C2212yescomplemented
27C31×#135—22no—
28C2×M16×C31C2211yescharacteristic
29#11758528—21no—
30#11758527—22no—
31#11758526—21no—
32#11758521C2212yescomplemented
33#11758519C2212yescomplemented
34#11758513C2212yescomplemented
35#11758512—22no—
36#11758509C2212yescomplemented
37#11758503C2211yescharacteristic
38#11758501—22no—
39C2×#92474C2211yescharacteristic
40#11758425C2214yescomplemented
41#11758413C2214yescomplemented
42#11758412C2214yescomplemented
43#41×C31—21no—
44#41×C31—21no—
45#41×C31—21no—
46#41×C31—21no—
47#41×C31—21no—
48#41×C31—21no—
49#41×C31D412yes—
50#41×C31C2312yes—
51C2×Q8×C31D411yescharacteristic
52C2×Q8×C31C2311yescharacteristic
53C2×D4×C31—21no—
54C2×D4×C31—21no—
55C2×D4×C31D411yescharacteristic
56C2×D4×C31C2311yescharacteristic
57C22×C4×C31—21no—
58C22×C4×C31—21no—
59C22×C4×C31C2311yescharacteristic
60C2×C22⋊D31—22no—
61C2×C22⋊D31—21no—
62C22×Dic31—22no—
63C22×Dic31D412yes—
64Q8×D31—24no—
65Q8×D31—24no—
66#92497—28no—
67#92497—24no—
68#92497—24no—
69#92497—28no—
70#92495—24no—
71#92495—21no—
72#92495—21no—
73#92495—24no—
74#92495D412yes—
75#92495C2312yes—
76C2×D124D411yescharacteristic
77C2×C4×D31—22no—
78C2×C4×D31—21no—
79C2×Dic62—22no—
80C2×Dic62—24no—
81C2×Dic62—21no—
82C2×Dic62—21no—
83C2×Dic62—21no—
84C2×Dic62—21no—
85C2×Dic62—24no—
86C2×Dic62D412yes—
87C2×Dic62D411yescharacteristic
88C2×Dic62C2312yes—
89Q16×C31—22no—
90SD16×C31—22no—
91SD16×C31—22no—
92D8×C31—22no—
93M16×C31—21no—
94M16×C31C2312yes—
95C2×C8×C31—21no—
96C22⋊Dic31—21no—
97C22⋊Dic31—24no—
98#92482—22no—
99#92481—22no—
100#92480—22no—
101#92479—22no—
102C4⋊Dic31—21no—
103Dic31⋊C4—24no—
104C4×Dic31C2312yes—
105#92474—21no—
106#92474C2312yes—
107C2×C31⋊C8—21no—
108C23×C31—21no—
109C23×C31—21no—
110C23×C31C2×D411yescharacteristic
111C22×D31—21no—
112Q8×C31—21no—
113Q8×C31—21no—
114Q8×C31—21no—
115Q8×C31C2×D412yes—
116D4×C31—22no—
117D4×C31—22no—
118D4×C31—21no—
119D4×C31—21no—
120D4×C31—21no—
121D4×C31—41no—
122D4×C31—41no—
123D4×C31C2×D412yes—
124C2×C4×C31—21no—
125C2×C4×C31—22no—
126C2×C4×C31—22no—
127C2×C4×C31—21no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2×D411yescharacteristic
135C2×C4×C31C2411yescharacteristic
136C22⋊D31—28no—
137C22⋊D31—22no—
138C22⋊D31—22no—
139C2×Dic31—28no—
140C2×Dic31—22no—
141C2×Dic31—21no—
142C2×Dic31—22no—
143C2×Dic31—22no—
144C2×Dic31C2×D414yes—
145D124—21no—
146D124C2×D412yes—
147C4×D31—28no—
148C4×D31—22no—
149C4×D31—22no—
150Dic62—28no—
151Dic62—22no—
152Dic62—21no—
153Dic62—22no—
154Dic62—22no—
155Dic62—22no—
156Dic62C2×D412yes—
157C8×C31—22no—
158C31⋊C8—22no—
159#3020—311nomaximal, Sylow
160C22×C31—22no—
161C22×C31—41no—
162C22×C31—42no—
163C22×C31—42no—
164C22×C31—41no—
165C22×C31#12011yescharacteristic
166C22×C31#12011yescharacteristic
167C22×C31C22×D411yescharacteristic
168C2×D31—22no—
169C2×D31—42no—
170C4×C31—22no—
171C4×C31—41no—
172C4×C31#12011yescharacteristic
173C4×C31#12011yescharacteristic
174C4×C31C22×D411yescharacteristic
175C4×C31C22×D411yescharacteristic
176Dic31—24no—
177Dic31—22no—
178Dic31—42no—
179C2×2−1+4—311no—
180#575—311no—
181#575D3111yescharacteristic, complemented
182#536—311no—
183#456—314no—
184#454—314no—
185#421—311no—
186#421—311no—
187C2×#101—311no—
188C2×C31—21no—
189C2×C31—21no—
190C2×C31—21no—
191C2×C31—42no—
192C2×C31—81no—
193C2×C31C2×#12011yescharacteristic
194D31—42no—
1952−1+4—314no—
1962−1+4—622no—
1972−1+4—622no—
1982−1+4—622no—
199C2×#41—311no—
200C2×#41—621no—
201C2×#41—622no—
202C2×#41—621no—
203C2×#41C2×D3111yescharacteristic
204C2×#41C2×D3111yescharacteristic
205C22×Q8—311no—
206C22×Q8—621no—
207C22×Q8—621no—
208#137—21no—
209#137—312no—
210#137—621no—
211#137C2×D3112yescomplemented
212AGL(1,ℤ/8)—21no—
213AGL(1,ℤ/8)—312no—
214AGL(1,ℤ/8)—621no—
215AGL(1,ℤ/8)C2×D3112yescomplemented
216#135—22no—
217#135—622no—
218C2×M16—311no—
219C2×M16C2×D3111yescharacteristic
220C4⋊Q8—312no—
221#124—312no—
222#123—622no—
223C22⋊Q8—622no—
224#117—311no—
225#104—314no—
226#104—314no—
227#101—314no—
228C31#302011yescharacteristic, complemented, Sylow
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—21no—
234#41—21no—
235#41—312no—
236#41—312no—
237#41—624no—
238#41—624no—
239#41—628no—
240#41—628no—
241#41—624no—
242#41—624no—
243#41—621no—
244#41—621no—
245#41C22⋊D3112yes—
246#41C22×D3112yes—
247C2×Q8—312no—
248C2×Q8—312no—
249C2×Q8—311no—
250C2×Q8—624no—
251C2×Q8—622no—
252C2×Q8—624no—
253C2×Q8—624no—
254C2×Q8—624no—
255C2×Q8—621no—
256C2×Q8—621no—
257C2×Q8—621no—
258C2×Q8—621no—
259C2×Q8C22⋊D3111yescharacteristic
260C2×Q8C22×D3111yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—311no—
264C2×D4—622no—
265C2×D4—621no—
266C2×D4C22⋊D3111yescharacteristic
267C2×D4C22×D3111yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—312no—
271C22×C4—622no—
272C22×C4—622no—
273C22×C4—621no—
274C22×C4C22×D3111yescharacteristic
275Q16—22no—
276Q16—622no—
277SD16—22no—
278SD16—22no—
279SD16—622no—
280SD16—622no—
281D8—22no—
282D8—622no—
283M16—21no—
284M16—312no—
285M16—621no—
286M16C22×D3112yes—
287C2×C8—21no—
288C2×C8—621no—
289C4⋊C4—624no—
290C4⋊C4—621no—
291C22⋊C4—624no—
292C22⋊C4—621no—
293C42—312no—
294C23—21no—
295C23—21no—
296C23—621no—
297C23D4×D3111yescharacteristic
298Q8—21no—
299Q8—21no—
300Q8—21no—
301Q8—312no—
302Q8—628no—
303Q8—622no—
304Q8—622no—
305Q8—621no—
306Q8—622no—
307Q8—622no—
308Q8C2×C22⋊D3112yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—41no—
315D4—41no—
316D4—312no—
317D4—628no—
318D4—622no—
319D4—622no—
320D4—621no—
321D4C2×C22⋊D3112yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—314no—
329C2×C4—628no—
330C2×C4—628no—
331C2×C4—622no—
332C2×C4—622no—
333C2×C4—621no—
334C2×C4—622no—
335C2×C4—622no—
336C2×C4—622no—
337C2×C4D4×D3111yescharacteristic
338C2×C4D4×D3111yescharacteristic
339C2×C4D4×D3111yescharacteristic
340C2×C4C2×C22⋊D3111yescharacteristic
341C2×C4C2×C22⋊D3111yescharacteristic
342C2×C4C23×D3111yescharacteristic
343C8—22no—
344C8—622no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—622no—
351C22—1242no—
352C22#1175851411yescharacteristic
353C22#1175851411yescharacteristic
354C22C2×D4×D3111yescharacteristic
355C4—22no—
356C4—41no—
357C4—624no—
358C4—622no—
359C4—1242no—
360C4#1175851411yescharacteristic
361C4#1175851411yescharacteristic
362C4C2×D4×D3111yescharacteristic
363C4C22×C22⋊D3111yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—1242no—
370C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#575C2
socleC2×C31C2×#120
hypercenter#575D31
hypocenterC31#3020
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#11758567C2semidirect
2. #14521847C2semidirect
3. #14521620C2semidirect
4. #14521230C2semidirect
5. #14521220C2semidirect
6. #14521066C2semidirect
7. #14521065C2semidirect
8. C2×#11758412C2semidirect
9. C31×#575C2semidirect
24. C31×#137C22semidirect
26. C31×AGL(1,ℤ/8)C22semidirect
32. #11758521C22semidirect
33. #11758519C22semidirect
34. #11758513C22semidirect
36. #11758509C22semidirect
40. #11758425C22semidirect
41. #11758413C22semidirect
42. #11758412C22semidirect
181. #575D31semidirect
211. #137C2×D31semidirect
215. AGL(1,ℤ/8)C2×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
#145218472
#145216202
#145212302
#145212202
#145210662
#145210652
C2×#117584122
C31×#5752
#302031