Finite Group Databasebeta

← #36363024

Subgroups — #36363024

Order 22144.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19149705C211yesmaximal, characteristic, complemented
2#26012037C211yesmaximal, characteristic, complemented
3C2×D4×Dic173C211yesmaximal, characteristic, complemented
4#26011708C212yesmaximal, complemented
5#26011704C212yesmaximal, complemented
6#26011694C212yesmaximal, complemented
7#26011684C212yesmaximal, complemented
8#26011514C211yesmaximal, characteristic, complemented
9#26011493C211yesmaximal, characteristic, complemented
10#26011492C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C23×Dic173—21no—
14#19149705—24no—
15C2×#14583814C2212yes—
16C2×#14583811C2212yes—
17C2×#14583810C2211yescharacteristic
18#19149677C2211yescharacteristic, complemented
19#19149675C2211yescharacteristic, complemented
20#19149668C2211yescharacteristic, complemented
21#19149666C2211yescharacteristic, complemented
22#19149651—21no—
23#19149651—21no—
24D4×Dic173—21no—
25D4×Dic173—21no—
26D4×Dic173C2212yes—
27D4×Dic173C2212yes—
28C4×C22⋊D173—21no—
29C4×C22⋊D173—21no—
30C2×C4⋊Dic173C2211yescharacteristic
31C2×C22⋊Dic173—21no—
32C2×C4×Dic173C2211yescharacteristic
33#19149608C2212yescomplemented
34#19149608C2212yescomplemented
35#19149604C2212yescomplemented
36#19149603C2212yescomplemented
37#19149601C2212yescomplemented
38#19149600C2212yescomplemented
39C2×#41×C173C2211yescharacteristic
40C22×D4×C173C2211yescharacteristic
41AGL(1,ℤ/8)×C173—24no—
42C2×SD16×C173C2212yes—
43C2×D8×C173C2212yes—
44C2×M16×C173C2211yescharacteristic
45#14583826—22no—
46#14583826—22no—
47C2×Dic346C2311yescharacteristic
48C2×D692C2311yescharacteristic
49C2×C22⋊D173—21no—
50C2×C22⋊D173—21no—
51C2×C22⋊D173—21no—
52C22×Dic173—22no—
53C22×Dic173—22no—
54C22×Dic173—21no—
55C22×Dic173—44no—
56C22×Dic173D411yescharacteristic
57C22×Dic173D411yescharacteristic
58C2×C4×D173—21no—
59#14583814—24no—
60#14583811—24no—
61#14583810—22no—
62C2×#13063436C2312yes—
63C4⋊Dic173—21no—
64C4⋊Dic173C2312yes—
65#14583805—21no—
66#14583805—21no—
67C22⋊Dic173—22no—
68C22⋊Dic173—22no—
69C22⋊Dic173—21no—
70C22⋊Dic173—21no—
71#14583803—21no—
72#14583803—21no—
73C4×Dic173D411yescharacteristic
74C4×Dic173D411yescharacteristic
75C4×Dic173C2311yescharacteristic
76C4×Dic173C2311yescharacteristic
77C24×C173—21no—
78#41×C173—22no—
79#41×C173—22no—
80C2×Q8×C173C2311yescharacteristic
81C2×D4×C173—21no—
82C2×D4×C173—21no—
83C2×D4×C173—22no—
84C2×D4×C173D414yescomplemented
85C2×D4×C173C2311yescharacteristic
86C2×D4×C173C2312yes—
87C22×C4×C173—21no—
88C22×C4×C173C2311yescharacteristic
89SD16×C173—24no—
90D8×C173—24no—
91M16×C173—22no—
92C2×C8×C173C2312yes—
93C22×D173—21no—
94Dic346—21no—
95Dic346—21no—
96D692—21no—
97D692—21no—
98C22⋊D173—42no—
99C22⋊D173—42no—
100C22⋊D173—42no—
101C2×Dic173—22no—
102C2×Dic173—21no—
103C2×Dic173—22no—
104C2×Dic173—41no—
105C2×Dic173—44no—
106C2×Dic173—42no—
107C2×Dic173—44no—
108C2×Dic173C2×D411yescharacteristic
109C2×Dic173C2×D411yescharacteristic
110C2×Dic173C2×D411yescharacteristic
111C2×Dic173C2×D411yescharacteristic
112C4×D173—42no—
113#13063436—22no—
114C23×C173—24no—
115C23×C173—22no—
116C23×C173—21no—
117C23×C173—42no—
118C23×C173—42no—
119C23×C173C2×D411yescharacteristic
120Q8×C173—21no—
121Q8×C173—21no—
122D4×C173—24no—
123D4×C173—21no—
124D4×C173—22no—
125D4×C173—21no—
126D4×C173—42no—
127D4×C173C2×D414yes—
128C2×C4×C173—21no—
129C2×C4×C173—42no—
130C2×C4×C173C2×D411yescharacteristic
131C2×C4×C173C2×D412yes—
132C2×C4×C173#4112yes—
133C2×C4×C173C2411yescharacteristic
134C8×C173—22no—
135C22×C173—21no—
136C22×C173—21no—
137C22×C173—24no—
138C22×C173—48no—
139C22×C173—42no—
140C22×C173—42no—
141C22×C173—41no—
142C22×C173—41no—
143C22×C173—41no—
144C22×C173#12112yes—
145C22×C173C22×D411yescharacteristic
146C2×D173—41no—
147C2×D173—41no—
148C2×D173—41no—
149C4×C173—41no—
150C4×C173#12112yes—
151C4×C173C22×D411yescharacteristic
152C4×C173C2×#4111yescharacteristic
153Dic173—21no—
154Dic173—21no—
155Dic173—21no—
156Dic173—41no—
157Dic173—42no—
158Dic173AGL(1,ℤ/8)12yescomplemented
159C2×C173—22no—
160C2×C173—44no—
161C2×C173—81no—
162C2×C173C2×#12111yescharacteristic
163C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C173#57511yescharacteristic
165D173—81no—
166C173#303611yescharacteristic, complemented, Sylow
167#3036—1731nomaximal, Sylow
168C2×AGL(1,ℤ/8)—1731no—
169C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
170#532—1731no—
171C2×C4×D4—1731no—
172#463—1732no—
173#463—1732no—
174#461—1732no—
175#459—1732no—
176#423—1731no—
177#417—1731no—
178#417—1731no—
179C2×#41—1731no—
180C2×#41C2×D17311yescharacteristic
181C22×D4C2×D17311yescharacteristic
182C23×C4—3461no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—3464no—
185C2×SD16—1732no—
186C2×SD16C2×D17312yes—
187C2×D8—1732no—
188C2×D8C2×D17312yes—
189C2×M16—1731no—
190C2×M16C2×D17311yescharacteristic
191C4⋊Q8—1731no—
192#127—1731no—
193#124—1731no—
194#124—1731no—
195#121—3461no—
196#121—3461no—
197C4×D4—3461no—
198C4×D4—3461no—
199C4×D4—3461no—
200C4×D4—3461no—
201C4×D4—1732no—
202C4×D4—1732no—
203C2×C4⋊C4—1731no—
204C2×C22⋊C4—3461no—
205C2×C42—1731no—
206C4⋊C8—1732no—
207C4⋊C8—1732no—
208#103—1732no—
209#103—1732no—
210#102—1732no—
211#102—1732no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—3462no—
216#41—3462no—
217C2×Q8—1731no—
218C2×Q8C22×D17311yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—3461no—
223C2×D4—3461no—
224C2×D4—3461no—
225C2×D4—1731no—
226C2×D4C22⋊D17314yescomplemented
227C2×D4C22×D17311yescharacteristic
228C2×D4C22×D17312yes—
229C22×C4—21no—
230C22×C4—3461no—
231C22×C4—3462no—
232C22×C4—3462no—
233C22×C4—3461no—
234C22×C4—1731no—
235C22×C4—1731no—
236C22×C4—6924no—
237C22×C4C22×D17311yescharacteristic
238SD16—24no—
239SD16—3464no—
240D8—24no—
241D8—3464no—
242M16—22no—
243M16—3462no—
244C2×C8—1732no—
245C2×C8C22×D17312yes—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—1732no—
250C22⋊C4—3461no—
251C22⋊C4—3461no—
252C22⋊C4—3462no—
253C22⋊C4—3462no—
254C22⋊C4—3461no—
255C22⋊C4—3461no—
256C42—1731no—
257C42—1731no—
258C42—1731no—
259C42—1731no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—3461no—
266C23D4×D17311yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—3461no—
270Q8—3461no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—3461no—
277D4—3461no—
278D4—6922no—
279D4—6922no—
280D4—6922no—
281D4C2×C22⋊D17314yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—3462no—
285C2×C4—3461no—
286C2×C4—3462no—
287C2×C4—1731no—
288C2×C4—1731no—
289C2×C4—1731no—
290C2×C4—1731no—
291C2×C4—6921no—
292C2×C4—6924no—
293C2×C4—6924no—
294C2×C4—6922no—
295C2×C4—6922no—
296C2×C4D4×D17311yescharacteristic
297C2×C4C2×C22⋊D17312yes—
298C2×C4#1458382512yes—
299C2×C4C23×D17311yescharacteristic
300C8—22no—
301C8—3462no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—6921no—
312C22—6921no—
313C22—6921no—
314C22#1914965112yes—
315C22C2×D4×D17311yescharacteristic
316C4—41no—
317C4—3461no—
318C4—3461no—
319C4—3461no—
320C4—1732no—
321C4—6921no—
322C4—6922no—
323C4#1914965112yes—
324C4C22×C22⋊D17311yescharacteristic
325C4C2×#1458382511yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—13841no—
330C2C2×#1914965111yescharacteristic
331C2AGL(1,ℤ/8)×D17311yescharacteristic
332C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3036
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. #26012037C2semidirect
3. C2×D4×Dic173C2semidirect
4. #26011708C2semidirect
5. #26011704C2semidirect
6. #26011694C2semidirect
7. #26011684C2semidirect
8. #26011514C2semidirect
9. #26011493C2semidirect
10. #26011492C2semidirect
11. C2×AGL(1,ℤ/8)×C173C2semidirect
18. #19149677C22semidirect
19. #19149675C22semidirect
20. #19149668C22semidirect
21. #19149666C22semidirect
33. #19149608C22semidirect
34. #19149608C22semidirect
35. #19149604C22semidirect
36. #19149603C22semidirect
37. #19149601C22semidirect
38. #19149600C22semidirect
84. C2×D4×C173D4semidirect
158. Dic173AGL(1,ℤ/8)semidirect
166. C173#3036semidirect
169. C2×AGL(1,ℤ/8)D173semidirect
226. C2×D4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
#260120372
C2×D4×Dic1732
#260117082
#260117042
#260116942
#260116842
#260115142
#260114932
#260114922
C2×AGL(1,ℤ/8)×C1732
#3036173