Finite Group Databasebeta

← #37543082

Subgroups — #37543082

Order 23168.

698 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 133 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
12−1+4×D181C211yesmaximal, characteristic, complemented
2#26042075C212yesmaximal, complemented
3#26041849C211yesmaximal, characteristic, complemented
4#26041464C211yesmaximal, characteristic, complemented
5#26041463C212yesmaximal, complemented
6#26041462C211yesmaximal, characteristic, complemented
7#26041451C211yesmaximal, characteristic, complemented
8#26041450C212yesmaximal, complemented
9#26041294C212yesmaximal, complemented
10#101×D181C211yesmaximal, characteristic, complemented
11#454×C181C211yesmaximal, characteristic, complemented
12#19153864—22no—
13#19153864C2211yescharacteristic
14#19153863—22no—
15#19153863—22no—
16#19153863C2212yes—
17C2×#15668448C2211yescharacteristic
18#41×D181—22no—
19#41×D181—22no—
20#41×D181C2212yes—
21C2×Q8×D181—22no—
22C2×Q8×D181C2211yescharacteristic
23#19153849C2212yescomplemented
24#19153846C2212yescomplemented
25#19153845—22no—
26#19153842C2212yescomplemented
27#19153840—22no—
28#19153837—22no—
29#19153835—22no—
30M16×D181C2212yes—
31#19153816C2211yescharacteristic, complemented
32#19153814C2211yescharacteristic, complemented
33#19153803C2211yescharacteristic, complemented
34#19153798—22no—
35#19153793—22no—
36#19153773C2211yescharacteristic
37#19153744C2212yescomplemented
38#19153743C2212yescomplemented
39#19153742C2212yescomplemented
40#19153735C2211yescharacteristic, complemented
41#19153734C2212yescomplemented
422−1+4×C181C2211yescharacteristic
43AGL(1,ℤ/8)×C181C2212yescomplemented
44#124×C181C2211yescharacteristic, complemented
45#104×C181C2212yescomplemented
46#101×C181C2211yescharacteristic, complemented
47#15668450—22no—
48#15668450—22no—
49#15668450D412yes—
50#15668449—24no—
51#15668449—24no—
52#15668449—22no—
53#15668449—22no—
54#15668449—22no—
55#15668449C2312yes—
56#15668448—24no—
57#15668448—22no—
58#15668448—24no—
59#15668448—22no—
60#15668448—22no—
61#15668448—22no—
62#15668448—22no—
63#15668448D412yes—
64C2×Dic362—24no—
65C2×Dic362—22no—
66C2×Dic362D412yes—
67C2×Dic362C2311yescharacteristic
68C2×Dic362C2311yescharacteristic
69Q8×D181—22no—
70Q8×D181—24no—
71Q8×D181—22no—
72Q8×D181—22no—
73Q8×D181—22no—
74Q8×D181—22no—
75Q8×D181D412yes—
76C2×C22⋊D181—21no—
77D4×D181—22no—
78D4×D181—22no—
79D4×D181D412yes—
80C22×Dic181—21no—
81C2×C4×D181—24no—
82C2×C4×D181—22no—
83C2×C4×D181D412yes—
84C2×C4×D181C2311yescharacteristic
85Dic724—22no—
86#15668439—22no—
87#15668438—22no—
88#15668437—22no—
89#15668437—22no—
90#15668434—22no—
91#15668433C2312yes—
92#15668432—22no—
93C8×D181—22no—
94C4⋊Dic181—22no—
95#15668428—21no—
96#15668428—22no—
97C22⋊Dic181—22no—
98#15668426—21no—
99C4×Dic181C2311yescharacteristic
100#41×C181—22no—
101#41×C181—22no—
102#41×C181C2312yes—
103C2×Q8×C181—22no—
104C2×Q8×C181C2311yescharacteristic
105C2×D4×C181C2311yescharacteristic
106SD16×C181—22no—
107D8×C181—22no—
108M16×C181C2312yes—
109C22⋊C4×C181—22no—
110C42×C181C2311yescharacteristic
111C22×D181—22no—
112C22×D181C2×D411yescharacteristic
113Dic362—22no—
114Dic362—24no—
115Dic362—22no—
116Dic362—24no—
117Dic362—22no—
118Dic362—22no—
119Dic362C2×D412yes—
120D724—22no—
121D724—22no—
122D724C2×D412yes—
123C22⋊D181—24no—
124C22⋊D181—22no—
125C22⋊D181—22no—
126C22⋊D181—22no—
127C22⋊D181—41no—
128C22⋊D181—41no—
129C2×Dic181—24no—
130C2×Dic181—22no—
131C2×Dic181—21no—
132C2×Dic181—21no—
133C2×Dic181—42no—
134C2×Dic181C2×D412yes—
135C2×Dic181C2×D411yescharacteristic
136C4×D181—24no—
137C4×D181—24no—
138C4×D181—22no—
139C4×D181—22no—
140C4×D181—22no—
141C4×D181—22no—
142C4×D181—22no—
143C4×D181C2×D412yes—
144C4×D181C2×D412yes—
145#13089678—22no—
146C23×C181—21no—
147Q8×C181—22no—
148Q8×C181—22no—
149Q8×C181C2×D412yes—
150D4×C181—22no—
151D4×C181—22no—
152D4×C181—22no—
153D4×C181C2×D412yes—
154C2×C4×C181—24no—
155C2×C4×C181—22no—
156C2×C4×C181—21no—
157C2×C4×C181C2×D412yes—
158C2×C4×C181C2411yescharacteristic
159C8×C181—22no—
160C22×C181—22no—
161C22×C181—42no—
162C22×C181—41no—
163C22×C181C22×D411yescharacteristic
164C2×D181—22no—
165C2×D181—41no—
166C2×D181—44no—
167C2×D181#12011yescharacteristic
168C2×D181#12011yescharacteristic
169C4×C181—22no—
170C4×C181—22no—
171C4×C181—41no—
172C4×C181C22×D412yes—
173Dic181—22no—
174Dic181—22no—
175Dic181—41no—
176Dic181—41no—
177Dic181#12012yes—
178C2×C181—21no—
179C2×C181—42no—
180C2×C181—81no—
181C2×C181C2×#12011yescharacteristic
182D181—21no—
183D181—21no—
184D181—42no—
185C181#302011yescharacteristic, complemented, Sylow
186#3020—1811nomaximal, Sylow
187C2×2−1+4—1811no—
188#575—1812no—
189#536—1811no—
190#456—1811no—
191#456—1812no—
192#456—1811no—
193#454—1812no—
194#454—1811no—
195#454D18111yescharacteristic, complemented
196#421—1812no—
197C2×#101—1811no—
1982−1+4—3622no—
1992−1+4—3622no—
2002−1+4—3622no—
2012−1+4—1812no—
2022−1+4—1811no—
2032−1+4C2×D18111yescharacteristic
204C2×#41—3622no—
205C2×#41—3622no—
206C2×#41—1812no—
207C2×#41—1811no—
208C22×Q8—3622no—
209C22×Q8—1811no—
210#137—3622no—
211#137—1812no—
212#137—1812no—
213AGL(1,ℤ/8)—3622no—
214AGL(1,ℤ/8)—1812no—
215AGL(1,ℤ/8)C2×D18112yescomplemented
216#135—3622no—
217#135—3622no—
218C2×M16—1812no—
219C4⋊Q8—1811no—
220C4⋊Q8—1811no—
221#124—1811no—
222#124C2×D18111yescharacteristic, complemented
223#123—3622no—
224C22⋊Q8—3622no—
225#117—1811no—
226#104—1812no—
227#104—1812no—
228#104—1812no—
229#104C2×D18112yescomplemented
230#101—1811no—
231#101—1812no—
232#101C2×D18111yescharacteristic, complemented
233#41—22no—
234#41—22no—
235#41—3624no—
236#41—3624no—
237#41—3622no—
238#41—3622no—
239#41—3624no—
240#41—3622no—
241#41—3622no—
242#41—3622no—
243#41—3622no—
244#41—3624no—
245#41—3622no—
246#41—3622no—
247#41—3622no—
248#41—3622no—
249#41—1812no—
250#41—1812no—
251#41—1812no—
252#41C22×D18112yes—
253C2×Q8—22no—
254C2×Q8—3624no—
255C2×Q8—3622no—
256C2×Q8—3622no—
257C2×Q8—3624no—
258C2×Q8—3622no—
259C2×Q8—3622no—
260C2×Q8—3622no—
261C2×Q8—3622no—
262C2×Q8—1812no—
263C2×Q8—1811no—
264C2×Q8—1812no—
265C2×Q8—1811no—
266C2×Q8C22×D18111yescharacteristic
267C2×D4—3622no—
268C2×D4—3622no—
269C2×D4—3621no—
270C2×D4—1812no—
271C2×D4C22×D18111yescharacteristic
272C22×C4—3624no—
273C22×C4—3622no—
274C22×C4—3621no—
275C22×C4—1812no—
276C22×C4—1811no—
277Q16—3622no—
278Q16—3622no—
279SD16—22no—
280SD16—3622no—
281SD16—3622no—
282SD16—3622no—
283D8—22no—
284D8—3622no—
285M16—3622no—
286M16—1812no—
287M16C22×D18112yes—
288C2×C8—3622no—
289C4⋊C4—3621no—
290C4⋊C4—3622no—
291C4⋊C4—3622no—
292C22⋊C4—22no—
293C22⋊C4—3621no—
294C22⋊C4—3622no—
295C42—1811no—
296C42C22×D18111yescharacteristic
297C23—21no—
298C23—3622no—
299C23—1811no—
300Q8—22no—
301Q8—22no—
302Q8—3624no—
303Q8—3622no—
304Q8—3622no—
305Q8—3624no—
306Q8—3622no—
307Q8—3622no—
308Q8—1812no—
309Q8D4×D18112yes—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—3622no—
314D4—3622no—
315D4—3624no—
316D4—3622no—
317D4—3622no—
318D4—3622no—
319D4—1812no—
320D4—7241no—
321D4—7241no—
322D4D4×D18112yes—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—3624no—
327C2×C4—3624no—
328C2×C4—3622no—
329C2×C4—3622no—
330C2×C4—3622no—
331C2×C4—3622no—
332C2×C4—3624no—
333C2×C4—3622no—
334C2×C4—3622no—
335C2×C4—3621no—
336C2×C4—3621no—
337C2×C4—1812no—
338C2×C4—1811no—
339C2×C4—1812no—
340C2×C4—1812no—
341C2×C4—7242no—
342C2×C4D4×D18112yes—
343C2×C4C23×D18111yescharacteristic
344C8—22no—
345C8—3622no—
346C22—22no—
347C22—42no—
348C22—41no—
349C22—3622no—
350C22—1811no—
351C22—1811no—
352C22—7241no—
353C22—7244no—
354C22C2×D4×D18111yescharacteristic
355C4—22no—
356C4—22no—
357C4—41no—
358C4—3622no—
359C4—3622no—
360C4—1812no—
361C4—7241no—
362C4—7241no—
363C4C2×D4×D18112yes—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—3621no—
368C2—3621no—
369C2—7242no—
370C2#120×D18111yescharacteristic
subgroupG/H
centerC2#120×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#454×C181C2
socleC2×C181C2×#120
hypercenter#454D181
hypocenterC181#3020
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. 2−1+4×D181C2semidirect
2. #26042075C2semidirect
3. #26041849C2semidirect
4. #26041464C2semidirect
5. #26041463C2semidirect
6. #26041462C2semidirect
7. #26041451C2semidirect
8. #26041450C2semidirect
9. #26041294C2semidirect
10. #101×D181C2semidirect
11. #454×C181C2semidirect
23. #19153849C22semidirect
24. #19153846C22semidirect
26. #19153842C22semidirect
31. #19153816C22semidirect
32. #19153814C22semidirect
33. #19153803C22semidirect
37. #19153744C22semidirect
38. #19153743C22semidirect
39. #19153742C22semidirect
40. #19153735C22semidirect
41. #19153734C22semidirect
43. AGL(1,ℤ/8)×C181C22semidirect
44. #124×C181C22semidirect
45. #104×C181C22semidirect
46. #101×C181C22semidirect
185. C181#3020semidirect
195. #454D181semidirect
215. AGL(1,ℤ/8)C2×D181semidirect
222. #124C2×D181semidirect
229. #104C2×D181semidirect
232. #101C2×D181semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D1812
#260420752
#260418492
#260414642
#260414632
#260414622
#260414512
#260414502
#260412942
#101×D1812
#454×C1812
#3020181