Finite Group Databasebeta

← #36363061

Subgroups — #36363061

Order 22144.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic, complemented
2C2×D8×D173C211yesmaximal, characteristic, complemented
3#26012054C211yesmaximal, characteristic, complemented
4#26011963C211yesmaximal, characteristic, complemented
5#26011707C211yesmaximal, characteristic, complemented
6#26011706C211yesmaximal, characteristic, complemented
7#26011705C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011685C211yesmaximal, characteristic, complemented
10#26011684C211yesmaximal, characteristic, complemented
11#26011683C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#459×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149703—24no—
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic, complemented
22C2×D1384C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24C2×#14583811C2211yescharacteristic
25C2×#14583811C2211yescharacteristic, complemented
26D8×D173—24no—
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149675C2211yescharacteristic, complemented
30#19149667C2211yescharacteristic, complemented
31#19149666C2211yescharacteristic, complemented
32#19149651—21no—
33#19149651—21no—
34#19149647—21no—
35#19149647—21no—
36C4×D692C2211yescharacteristic, complemented
37D4×Dic173C2211yescharacteristic, complemented
38#19149641C2211yescharacteristic, complemented
39C4×C22⋊D173—21no—
40#19149639—21no—
41#19149637C2211yescharacteristic
42#19149636C2211yescharacteristic
43C2×#14583803—21no—
44C22⋊C4×D173—21no—
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149608C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149602C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149600C2211yescharacteristic, complemented
53#19149599C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic, complemented
56#127×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63#14583825—22no—
64#14583825—22no—
65C2×Dic346C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73D4×D173—22no—
74D4×D173—22no—
75D4×D173—22no—
76D4×D173D414yescomplemented
77C22×Dic173—21no—
78C22×Dic173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173C2311yescharacteristic
82#14583815—22no—
83#14583814—22no—
84D1384—22no—
85#14583811—22no—
86#14583811—22no—
87#14583811—22no—
88#14583809—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C8×D173—22no—
92C4⋊Dic173C2311yescharacteristic
93#14583805D411yescharacteristic
94#14583805D411yescharacteristic
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803D411yescharacteristic
104#14583803D411yescharacteristic
105C4×Dic173—21no—
106C4×Dic173C2311yescharacteristic
107C2×D4×C173—21no—
108C2×D4×C173—21no—
109C2×D4×C173C2311yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112C22×C4×C173—21no—
113D8×C173—22no—
114D8×C173—22no—
115C2×C8×C173C2311yescharacteristic
116C2×C8×C173C2311yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173—21no—
121C22×D173—24no—
122C22×D173—21no—
123C22×D173—42no—
124C22×D173—81no—
125C22×D173—81no—
126C22×D173—81no—
127C22×D173—81no—
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—21no—
131D692—21no—
132D692—21no—
133D692—21no—
134D692C2×D412yes—
135C22⋊D173—24no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—81no—
139C22⋊D173—81no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—42no—
143C2×Dic173—81no—
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C2×Dic173C2×D411yescharacteristic
147C4×D173—81no—
148C4×D173—81no—
149C4×D173C2×D412yes—
150C4×D173#4112yes—
151#13063436—21no—
152#13063436—21no—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—81no—
162D4×C173—81no—
163D4×C173C2×D412yes—
164C2×C4×C173—21no—
165C2×C4×C173—81no—
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×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173C22×D411yescharacteristic
181C2×D173—21no—
182C2×D173—21no—
183C2×D173—22no—
184C2×D173—41no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—41no—
189C2×D173—84no—
190C2×D173—84no—
191C2×D173#12112yes—
192C4×C173—21no—
193C4×C173—21no—
194C4×C173—41no—
195C4×C173C22×D411yescharacteristic
196C4×C173C2×#4111yescharacteristic
197Dic173—21no—
198Dic173—21no—
199Dic173—41no—
200Dic173—41no—
201Dic173#12112yes—
202C2×C173—42no—
203C2×C173—81no—
204C2×C173—81no—
205C2×C173C2×#12111yescharacteristic
206C2×C173#57611yescharacteristic
207C2×C173#57611yescharacteristic
208D173—22no—
209D173—42no—
210D173—81no—
211C173#304111yescharacteristic, complemented, Sylow
212#3041—1731nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1731no—
214C22×D8—1731no—
215#535—1731no—
216#518—1731no—
217#463—1731no—
218#463—1731no—
219#463—1731no—
220#463—1731no—
221#459—1731no—
222#459—1731no—
223#459—1731no—
224#459D17311yescharacteristic, complemented
225#424—1731no—
226#418—1731no—
227#416—1731no—
228C2×#41—1731no—
229C22×D4—1731no—
230C22×D4—1731no—
231AGL(1,ℤ/8)—3464no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×D8—3464no—
235C2×D8—1731no—
236C2×D8—1731no—
237C2×D8—1731no—
238C2×D8—1731no—
239C2×D8C2×D17311yescharacteristic
240C2×D8C2×D17311yescharacteristic, complemented
241C2×M16—1731no—
242C22×C8—1731no—
243#127—1731no—
244#127C2×D17311yescharacteristic, complemented
245#124—1731no—
246#124—1731no—
247#121—3461no—
248#121—3461no—
249#120—3461no—
250#120—3461no—
251C4×D4—3461no—
252C4×D4—3461no—
253C4×D4—1731no—
254C4×D4—1731no—
255C4×D4—1731no—
256C4×D4C2×D17311yescharacteristic, complemented
257#117—1731no—
258#117—1731no—
259C2×C22⋊C4—3461no—
260C2×C22⋊C4—3461no—
261C4⋊C8—1731no—
262C4⋊C8—1731no—
263C4⋊C8—1731no—
264C4⋊C8C2×D17311yescharacteristic, complemented
265#103—1731no—
266#103—1731no—
267#102—1731no—
268#102—1731no—
269#102—1731no—
270#102—1731no—
271#102C2×D17311yescharacteristic, complemented
272#102C2×D17311yescharacteristic, complemented
273C24—3461no—
274C24—3461no—
275#41—3462no—
276#41—3462no—
277C2×Q8—1731no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3461no—
281C2×D4—3462no—
282C2×D4—3462no—
283C2×D4—3462no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—3461no—
288C2×D4—1734no—
289C2×D4—1731no—
290C2×D4—1731no—
291C2×D4C22×D17311yescharacteristic
292C2×D4C22×D17311yescharacteristic
293C2×D4C22×D17311yescharacteristic
294C22×C4—21no—
295C22×C4—3461no—
296C22×C4—3461no—
297C22×C4—3461no—
298C22×C4—3461no—
299C22×C4—1731no—
300SD16—3462no—
301SD16—3462no—
302D8—22no—
303D8—22no—
304D8—3462no—
305D8—3462no—
306D8—3462no—
307D8—3462no—
308M16—3462no—
309C2×C8—3462no—
310C2×C8—1731no—
311C2×C8—1731no—
312C2×C8C22×D17311yescharacteristic
313C2×C8C22×D17311yescharacteristic
314C4⋊C4—1731no—
315C4⋊C4—1731no—
316C4⋊C4—1731no—
317C4⋊C4C22×D17311yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—1731no—
328C22⋊C4—1731no—
329C42—3461no—
330C42—1731no—
331C42C22×D17311yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—3464no—
336C23—3461no—
337C23—3461no—
338C23—13841no—
339C23—13841no—
340C23—13841no—
341C23—13841no—
342C23—1731no—
343C23—6922no—
344Q8—3461no—
345Q8—3461no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—3464no—
354D4—3461no—
355D4—3461no—
356D4—3461no—
357D4—13841no—
358D4—13841no—
359D4—1732no—
360D4—6922no—
361D4—6922no—
362D4D4×D17312yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—13841no—
368C2×C4—13841no—
369C2×C4—13841no—
370C2×C4—1731no—
371C2×C4—1731no—
372C2×C4—1731no—
373C2×C4—1732no—
374C2×C4—1732no—
375C2×C4—6922no—
376C2×C4D4×D17311yescharacteristic
377C2×C4D4×D17311yescharacteristic
378C2×C4C23×D17311yescharacteristic
379C8—21no—
380C8—21no—
381C8—3461no—
382C8—3461no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—3461no—
393C22—3461no—
394C22—3462no—
395C22—13844no—
396C22—13844no—
397C22—1732no—
398C22—6921no—
399C22—6921no—
400C22—6921no—
401C22—6921no—
402C22—6921no—
403C22C2×D4×D17311yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—3461no—
408C4—3461no—
409C4—1732no—
410C4—6921no—
411C4—6921no—
412C4C2×D4×D17311yescharacteristic
413C4#41×D17311yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—3462no—
418C2—13841no—
419C2—6922no—
420C2#121×D17311yescharacteristic
421C2#2601229211yescharacteristic
422C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3041
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×D8×D173C2semidirect
3. #26012054C2semidirect
4. #26011963C2semidirect
5. #26011707C2semidirect
6. #26011706C2semidirect
7. #26011705C2semidirect
8. #26011704C2semidirect
9. #26011685C2semidirect
10. #26011684C2semidirect
11. #26011683C2semidirect
12. #26011516C2semidirect
13. #26011494C2semidirect
14. #26011484C2semidirect
15. #459×C173C2semidirect
20. C2×#14583815C22semidirect
21. C2×#14583814C22semidirect
25. C2×#14583811C22semidirect
29. #19149675C22semidirect
30. #19149667C22semidirect
31. #19149666C22semidirect
36. C4×D692C22semidirect
37. D4×Dic173C22semidirect
38. #19149641C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149602C22semidirect
50. #19149601C22semidirect
51. #19149600C22semidirect
52. #19149600C22semidirect
53. #19149599C22semidirect
55. C2×D8×C173C22semidirect
56. #127×C173C22semidirect
57. C4×D4×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #102×C173C22semidirect
60. #102×C173C22semidirect
76. D4×D173D4semidirect
211. C173#3041semidirect
224. #459D173semidirect
240. C2×D8C2×D173semidirect
244. #127C2×D173semidirect
256. C4×D4C2×D173semidirect
264. C4⋊C8C2×D173semidirect
271. #102C2×D173semidirect
272. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×D8×D1732
#260120542
#260119632
#260117072
#260117062
#260117052
#260117042
#260116852
#260116842
#260116832
#260115162
#260114942
#260114842
#459×C1732
#3041173