Finite Group Databasebeta

← #36365170

Subgroups — #36365170

Order 22144.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149705C211yesmaximal, characteristic, complemented
2C2×#19149704C211yesmaximal, characteristic, complemented
3#26012137C211yesmaximal, characteristic, complemented
4#26011873C211yesmaximal, characteristic, complemented
5#26011739C211yesmaximal, characteristic, complemented
6#26011735C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011703C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011624C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14#26011569C211yesmaximal, characteristic, complemented
15#546×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic, complemented
18#19149705—24no—
19#19149704—24no—
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic, complemented
22C2×#14583814C2211yescharacteristic, complemented
23C2×#14583813C2211yescharacteristic, complemented
24C2×#14583813C2211yescharacteristic, complemented
25C2×#14583811C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583810C2211yescharacteristic, complemented
30C2×#14583810C2211yescharacteristic, complemented
31#19149667C2211yescharacteristic, complemented
32#19149664—21no—
33#19149658C2211yescharacteristic, complemented
34#19149653C2211yescharacteristic
35#19149652—21no—
36#19149647—21no—
37C4×D692C2211yescharacteristic, complemented
38C22⋊C4×D173—21no—
39#19149612C2211yescharacteristic, complemented
40C4⋊#13063436C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149602C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic, complemented
44#19149600C2211yescharacteristic, complemented
45#19149599C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47C22⋊#13063436C2211yescharacteristic, complemented
48C22⋊#13063436C2211yescharacteristic, complemented
49C4×#13063436C2211yescharacteristic, complemented
50C2×#41×C173C2211yescharacteristic
51C22×D4×C173C2211yescharacteristic, complemented
52#124×C173C2211yescharacteristic, complemented
53#123×C173—21no—
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173—21no—
56#121×C173C2211yescharacteristic
57#120×C173—21no—
58C4×D4×C173C2211yescharacteristic
59C4×D4×C173C2211yescharacteristic, complemented
60C2×C22⋊C4×C173—21no—
61C23×D173—21no—
62#14583826—22no—
63#14583826—41no—
64C2×Dic346C2311yescharacteristic
65C2×D692—22no—
66C2×D692—21no—
67C2×D692—41no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74#14583816—41no—
75#14583814—22no—
76#14583814—22no—
77#14583813—22no—
78#14583813—22no—
79#14583811—22no—
80#14583811—22no—
81#14583811—41no—
82#14583811D414yescomplemented
83#14583810—22no—
84#14583810—22no—
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C2×#13063436C2311yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C24×C173—21no—
97#41×C173—22no—
98#41×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—22no—
101C2×D4×C173—21no—
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173—41no—
105C2×D4×C173D411yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C2×D4×C173C2311yescharacteristic
108C22×C4×C173—21no—
109C22×C4×C173—21no—
110C22×C4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112C4⋊C4×C173—21no—
113C4⋊C4×C173D411yescharacteristic
114C4⋊C4×C173C2311yescharacteristic
115C22⋊C4×C173—21no—
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128Dic346—21no—
129Dic346—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—42no—
134D692C2×D412yes—
135C22⋊D173—42no—
136C22⋊D173—81no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C4×D173—42no—
140C4×D173—81no—
141#13063436—21no—
142#13063436—21no—
143#13063436—21no—
144#13063436C2×D412yes—
145C23×C173—21no—
146C23×C173—21no—
147C23×C173—42no—
148C23×C173—42no—
149C23×C173—81no—
150C23×C173C2×D411yescharacteristic
151C23×C173C2×D411yescharacteristic
152Q8×C173—21no—
153Q8×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—41no—
158D4×C173—42no—
159D4×C173—42no—
160D4×C173—81no—
161D4×C173C2×D412yes—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—42no—
169C2×C4×C173—81no—
170C2×C4×C173C2×D411yescharacteristic
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2×D411yescharacteristic
173C2×C4×C173C2×D411yescharacteristic
174C2×C4×C173C2411yescharacteristic
175C22×C173—21no—
176C22×C173—21no—
177C22×C173—22no—
178C22×C173—21no—
179C22×C173—21no—
180C22×C173—21no—
181C22×C173—21no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—84no—
188C22×C173C22×D411yescharacteristic
189C2×D173—22no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—41no—
195C2×D173—84no—
196C4×C173—21no—
197C4×C173—21no—
198C4×C173—21no—
199C4×C173—21no—
200C4×C173—41no—
201C4×C173—41no—
202C4×C173C22×D411yescharacteristic
203C4×C173H(5,𝔽2)11yescharacteristic
204Dic173—41no—
205Dic173—41no—
206C2×C173—41no—
207C2×C173—41no—
208C2×C173—42no—
209C2×C173—81no—
210C2×C173D4211yescharacteristic
211C2×C173#57511yescharacteristic
212C2×C173#57611yescharacteristic
213D173—42no—
214D173—81no—
215C173#327311yescharacteristic, complemented, Sylow
216#3273—1731nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1731no—
218C2×AGL(1,ℤ/8)—1731no—
219#546—1731no—
220#546D17311yescharacteristic, complemented
221#495—1731no—
222#468—1731no—
223#468—1731no—
224#463—1731no—
225#463—1731no—
226#452—1731no—
227#452—1731no—
228#447—1731no—
229#447—1731no—
230C4×D8—1731no—
231#436—1731no—
232C2×#41—1731no—
233C2×#41C2×D17311yescharacteristic
234C22×D4—1731no—
235C22×D4C2×D17311yescharacteristic, complemented
236AGL(1,ℤ/8)—3464no—
237AGL(1,ℤ/8)—3464no—
238C2×Q16—1731no—
239C2×SD16—1731no—
240C2×SD16—1731no—
241C2×SD16—1731no—
242C2×SD16—1731no—
243C2×D8—1731no—
244C2×D8—1731no—
245C2×D8—1731no—
246C2×D8—1731no—
247C2×M16—1731no—
248C2×M16—1731no—
249#124—1731no—
250#124C2×D17311yescharacteristic, complemented
251#123—21no—
252#123—3461no—
253C22⋊Q8—1731no—
254C22⋊Q8C2×D17311yescharacteristic, complemented
255#121—21no—
256#121—3461no—
257#121—1731no—
258#121C2×D17311yescharacteristic
259#120—21no—
260#120—3461no—
261C4×D4—1731no—
262C4×D4C2×D17311yescharacteristic
263C4×D4C2×D17311yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—3461no—
266#107—1731no—
267C4⋊C8—1731no—
268#103—1731no—
269#103—1731no—
270#102—1731no—
271#102—1731no—
272#102—1731no—
273#102—1731no—
274C22⋊C8—1731no—
275C22⋊C8—1731no—
276C4×C8—1731no—
277C24—21no—
278C24—3461no—
279#41—22no—
280#41—41no—
281#41—3462no—
282#41—6921no—
283C2×Q8—1731no—
284C2×Q8C22×D17311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—3462no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—3461no—
294C2×D4—1731no—
295C2×D4—1731no—
296C2×D4—6921no—
297C2×D4C22⋊D17311yescharacteristic
298C2×D4C22×D17311yescharacteristic
299C2×D4C22×D17311yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4C22×D17311yescharacteristic
305C22×C4C22×D17311yescharacteristic
306Q16—6921no—
307SD16—3462no—
308SD16—3462no—
309SD16—3462no—
310SD16—3462no—
311D8—3462no—
312D8—3462no—
313D8—1734no—
314D8—6921no—
315M16—3462no—
316M16—3462no—
317C2×C8—1731no—
318C2×C8—1731no—
319C2×C8—1731no—
320C2×C8—1731no—
321C4⋊C4—21no—
322C4⋊C4—3461no—
323C4⋊C4—1731no—
324C4⋊C4C22⋊D17311yescharacteristic
325C4⋊C4C22×D17311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—3461no—
335C22⋊C4—3461no—
336C22⋊C4C22⋊D17311yescharacteristic
337C22⋊C4C22⋊D17311yescharacteristic
338C42C22×D17311yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3461no—
345C23—3461no—
346C23—13841no—
347C23—6922no—
348C23—6922no—
349C23C2×C22⋊D17311yescharacteristic
350C23C2×C22⋊D17311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3461no—
354Q8—3461no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—3461no—
363D4—3461no—
364D4—3461no—
365D4—13841no—
366D4—1732no—
367D4—6922no—
368D4—6922no—
369D4D4×D17312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—3461no—
379C2×C4—3461no—
380C2×C4—13841no—
381C2×C4—6922no—
382C2×C4C2×C22⋊D17311yescharacteristic
383C2×C4C2×C22⋊D17311yescharacteristic
384C2×C4C2×C22⋊D17311yescharacteristic
385C2×C4C2×C22⋊D17311yescharacteristic
386C2×C4C23×D17311yescharacteristic
387C8—3461no—
388C8—3461no—
389C8—3461no—
390C8—1732no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3462no—
405C22—13844no—
406C22—6921no—
407C22—6921no—
408C22—6921no—
409C22—6921no—
410C22—6921no—
411C22C22×C22⋊D17311yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—6921no—
419C4—6921no—
420C4C2×D4×D17311yescharacteristic
421C4#1914972411yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—13841no—
427C2—6922no—
428C2D4×C22⋊D17311yescharacteristic
429C2#2601228911yescharacteristic
430C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#546×C173C2
socleC22×C173C22×D4
hypercenter#546D173
hypocenterC173#3273
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C2×#19149704C2semidirect
3. #26012137C2semidirect
4. #26011873C2semidirect
5. #26011739C2semidirect
6. #26011735C2semidirect
7. #26011706C2semidirect
8. #26011703C2semidirect
9. #26011653C2semidirect
10. #26011651C2semidirect
11. #26011625C2semidirect
12. #26011624C2semidirect
13. C4×#14583811C2semidirect
14. #26011569C2semidirect
15. #546×C173C2semidirect
17. C22×D692C22semidirect
20. C2×#14583816C22semidirect
21. C2×#14583814C22semidirect
22. C2×#14583814C22semidirect
23. C2×#14583813C22semidirect
24. C2×#14583813C22semidirect
29. C2×#14583810C22semidirect
30. C2×#14583810C22semidirect
31. #19149667C22semidirect
33. #19149658C22semidirect
37. C4×D692C22semidirect
39. #19149612C22semidirect
40. C4⋊#13063436C22semidirect
41. #19149603C22semidirect
42. #19149602C22semidirect
43. #19149600C22semidirect
44. #19149600C22semidirect
45. #19149599C22semidirect
46. #19149599C22semidirect
47. C22⋊#13063436C22semidirect
48. C22⋊#13063436C22semidirect
49. C4×#13063436C22semidirect
51. C22×D4×C173C22semidirect
52. #124×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
59. C4×D4×C173C22semidirect
82. #14583811D4semidirect
215. C173#3273semidirect
220. #546D173semidirect
235. C22×D4C2×D173semidirect
250. #124C2×D173semidirect
254. C22⋊Q8C2×D173semidirect
263. C4×D4C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C2×#191497042
#260121372
#260118732
#260117392
#260117352
#260117062
#260117032
#260116532
#260116512
#260116252
#260116242
C4×#145838112
#260115692
#546×C1732
#3273173