Finite Group Databasebeta

← #36363025

Subgroups — #36363025

Order 22144.

548 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149706C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3#26012040C211yesmaximal, characteristic, complemented
4C4×D4×D173C211yesmaximal, characteristic, complemented
5#26011707C211yesmaximal, characteristic, complemented
6#26011706C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011695C211yesmaximal, characteristic, complemented
9#26011694C211yesmaximal, characteristic, complemented
10#26011685C211yesmaximal, characteristic, complemented
11#26011684C211yesmaximal, characteristic, complemented
12#26011513C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14#26011489C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19C22×C4×D173—21no—
20#19149706—24no—
21#19149703—24no—
22C2×#14583815C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583813C2211yescharacteristic
25C2×D1384C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C2×#14583809C2211yescharacteristic
29C2×#14583809C2211yescharacteristic
30#19149677C2211yescharacteristic, complemented
31#19149675C2211yescharacteristic, complemented
32#19149667C2211yescharacteristic, complemented
33#19149652—21no—
34#19149652—21no—
35C4×D692C2211yescharacteristic
36D4×Dic173C2211yescharacteristic
37#19149641C2211yescharacteristic
38C4×C22⋊D173—21no—
39#19149639—21no—
40#19149639—21no—
41#19149639—21no—
42C4⋊C4×D173C2211yescharacteristic
43C22⋊C4×D173—21no—
44C42×D173C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149608C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#19149600C2211yescharacteristic, complemented
53#19149599C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—22no—
64#14583825—22no—
65#14583825—22no—
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74D4×D173—22no—
75D4×D173D414yescomplemented
76C22×Dic173—21no—
77C22×Dic173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—44no—
83C2×C4×D173D411yescharacteristic
84C2×C4×D173D411yescharacteristic
85C2×C4×D173C2311yescharacteristic
86#14583815—22no—
87#14583814—22no—
88#14583813—22no—
89D1384—22no—
90#14583811—22no—
91#14583811—22no—
92#14583809—22no—
93#14583809—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100C22⋊Dic173—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105C4×Dic173D411yescharacteristic
106C4×Dic173D411yescharacteristic
107C4×Dic173C2311yescharacteristic
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C22×C4×C173—21no—
112SD16×C173—22no—
113D8×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173C2311yescharacteristic
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C42×C173C2311yescharacteristic
121C22×D173—21no—
122C22×D173—24no—
123C22×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173C2×D411yescharacteristic
127Dic346—21no—
128Dic346—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—42no—
133D692C2×D412yes—
134C22⋊D173—24no—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—42no—
141C2×Dic173—42no—
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C4×D173—22no—
146C4×D173—41no—
147C4×D173—42no—
148C4×D173—42no—
149C4×D173—42no—
150C4×D173—42no—
151C4×D173C2×D412yes—
152C4×D173#4112yes—
153#13063436—21no—
154#13063436—21no—
155C23×C173—21no—
156C23×C173—21no—
157Q8×C173—21no—
158Q8×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173C2×D412yes—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—42no—
166C2×C4×C173C2×D411yescharacteristic
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C8×C173—21no—
170C8×C173—21no—
171C22×C173—22no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173C22×D411yescharacteristic
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—22no—
181C2×D173—48no—
182C2×D173—41no—
183C2×D173—41no—
184C2×D173—41no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173#12112yes—
188C4×C173—21no—
189C4×C173—21no—
190C4×C173—41no—
191C4×C173—41no—
192C4×C173C22×D411yescharacteristic
193C4×C173C2×#4111yescharacteristic
194Dic173—21no—
195Dic173—41no—
196Dic173—41no—
197Dic173#12112yes—
198Dic173AGL(1,ℤ/8)12yescomplemented
199C2×C173—42no—
200C2×C173—81no—
201C2×C173C2×#12111yescharacteristic
202C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C173#57511yescharacteristic
204D173—22no—
205D173—42no—
206D173—81no—
207C173#303611yescharacteristic, complemented, Sylow
208#3036—1731nomaximal, Sylow
209C2×AGL(1,ℤ/8)—1731no—
210C2×AGL(1,ℤ/8)—1731no—
211#532—1731no—
212C2×C4×D4—1731no—
213#463—1731no—
214#463—1731no—
215#463—1731no—
216#463D17311yescharacteristic, complemented
217#461—1731no—
218#461—1731no—
219#459—1731no—
220#459—1731no—
221#423—1731no—
222#417—1731no—
223#417—1731no—
224C2×#41—1731no—
225C2×#41—1731no—
226C22×D4—1731no—
227C23×C4—3461no—
228AGL(1,ℤ/8)—3464no—
229AGL(1,ℤ/8)—3464no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×SD16—1731no—
233C2×SD16C2×D17311yescharacteristic
234C2×D8—1731no—
235C2×D8—1731no—
236C2×D8—1731no—
237C2×D8C2×D17311yescharacteristic
238C2×M16—1731no—
239C2×M16—1731no—
240C4⋊Q8—1731no—
241#127—1731no—
242#124—1731no—
243#124C2×D17311yescharacteristic, complemented
244#121—3461no—
245#121—3461no—
246C4×D4—3461no—
247C4×D4—3461no—
248C4×D4—3461no—
249C4×D4—3461no—
250C4×D4—1731no—
251C4×D4—1731no—
252C4×D4—1731no—
253C4×D4C2×D17311yescharacteristic
254C2×C4⋊C4—1731no—
255C2×C22⋊C4—3461no—
256C2×C42—1731no—
257C4⋊C8—1731no—
258C4⋊C8—1731no—
259C4⋊C8—1731no—
260C4⋊C8C2×D17311yescharacteristic, complemented
261#103—1731no—
262#103—1731no—
263#103—1731no—
264#103C2×D17311yescharacteristic, complemented
265#102—1731no—
266#102—1731no—
267#102—1731no—
268#102C2×D17311yescharacteristic, complemented
269C24—3461no—
270#41—3462no—
271#41—3462no—
272#41—3462no—
273#41—3462no—
274C2×Q8—1731no—
275C2×Q8C22×D17311yescharacteristic
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—3462no—
279C2×D4—3461no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—1734no—
283C2×D4—1731no—
284C2×D4—1731no—
285C2×D4C22×D17311yescharacteristic
286C2×D4C22×D17311yescharacteristic
287C22×C4—21no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—3461no—
292C22×C4—3461no—
293C22×C4—3461no—
294C22×C4—1731no—
295C22×C4—1731no—
296C22×C4—1731no—
297C22×C4—6924no—
298SD16—22no—
299SD16—3462no—
300SD16—3462no—
301SD16—3462no—
302D8—22no—
303D8—3462no—
304D8—3462no—
305D8—3462no—
306M16—3462no—
307M16—3462no—
308C2×C8—1731no—
309C2×C8—1731no—
310C2×C8C22×D17311yescharacteristic
311C2×C8C22×D17311yescharacteristic
312C4⋊C4—3461no—
313C4⋊C4—3461no—
314C4⋊C4—3461no—
315C4⋊C4—1731no—
316C4⋊C4C22×D17311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C42—1731no—
326C42—1731no—
327C42—1731no—
328C42C22×D17311yescharacteristic
329C23—21no—
330C23—21no—
331C23—3464no—
332C23—3461no—
333C23—3461no—
334C23—1731no—
335C23—6922no—
336C23—6922no—
337Q8—21no—
338Q8—21no—
339Q8—3461no—
340Q8—3461no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—3464no—
345D4—3461no—
346D4—3461no—
347D4—3461no—
348D4—1732no—
349D4—6922no—
350D4—6922no—
351D4—6922no—
352D4—6922no—
353D4D4×D17312yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—3462no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—1731no—
361C2×C4—1731no—
362C2×C4—1731no—
363C2×C4—1732no—
364C2×C4—1732no—
365C2×C4—6921no—
366C2×C4—6922no—
367C2×C4—6922no—
368C2×C4—6922no—
369C2×C4—6922no—
370C2×C4—6922no—
371C2×C4—6922no—
372C2×C4D4×D17311yescharacteristic
373C2×C4D4×D17311yescharacteristic
374C2×C4C23×D17311yescharacteristic
375C8—21no—
376C8—21no—
377C8—3461no—
378C8—3461no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—3461no—
386C22—3461no—
387C22—3462no—
388C22—1732no—
389C22—6928no—
390C22—6921no—
391C22—6921no—
392C22—6921no—
393C22—6921no—
394C22—6921no—
395C22C2×D4×D17311yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—3461no—
401C4—1732no—
402C4—1732no—
403C4—6921no—
404C4—6921no—
405C4C2×D4×D17311yescharacteristic
406C4#41×D17311yescharacteristic
407C2—42no—
408C2—81no—
409C2—3462no—
410C2—13841no—
411C2—6922no—
412C2#121×D17311yescharacteristic
413C2AGL(1,ℤ/8)×D17311yescharacteristic
414C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
hypercenter#463D173
hypocenterC173#3036
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149703C2semidirect
3. #26012040C2semidirect
4. C4×D4×D173C2semidirect
5. #26011707C2semidirect
6. #26011706C2semidirect
7. #26011703C2semidirect
8. #26011695C2semidirect
9. #26011694C2semidirect
10. #26011685C2semidirect
11. #26011684C2semidirect
12. #26011513C2semidirect
13. #26011491C2semidirect
14. #26011489C2semidirect
15. #463×C173C2semidirect
30. #19149677C22semidirect
31. #19149675C22semidirect
32. #19149667C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149603C22semidirect
50. #19149602C22semidirect
51. #19149601C22semidirect
52. #19149600C22semidirect
53. #19149599C22semidirect
56. #124×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
75. D4×D173D4semidirect
198. Dic173AGL(1,ℤ/8)semidirect
207. C173#3036semidirect
216. #463D173semidirect
243. #124C2×D173semidirect
260. C4⋊C8C2×D173semidirect
264. #103C2×D173semidirect
268. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×#191497032
#260120402
C4×D4×D1732
#260117072
#260117062
#260117032
#260116952
#260116942
#260116852
#260116842
#260115132
#260114912
#260114892
#463×C1732
#3036173