Finite Group Databasebeta

← #36365167

Subgroups — #36365167

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×#19149706C211yesmaximal, characteristic, complemented
2C2×#19149704C211yesmaximal, characteristic, complemented
3#26012132C211yesmaximal, characteristic, complemented
4#26012129C211yesmaximal, characteristic, complemented
5#26011876C211yesmaximal, characteristic, complemented
6#26011740C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011709C211yesmaximal, characteristic, complemented
9#26011708C211yesmaximal, characteristic, complemented
10#26011652C211yesmaximal, characteristic, complemented
11#26011627C211yesmaximal, characteristic, complemented
12#26011626C211yesmaximal, characteristic, complemented
13#26011571C211yesmaximal, characteristic, complemented
14#26011566C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×D692C2211yescharacteristic, complemented
18C2×D4×D173C2211yescharacteristic, complemented
19#19149706—24no—
20#19149704—24no—
21C2×#14583813C2211yescharacteristic, complemented
22C2×#14583813C2211yescharacteristic, complemented
23C2×#14583813C2211yescharacteristic, complemented
24C2×D1384C2211yescharacteristic
25C2×D1384C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C2×#14583810C2211yescharacteristic, complemented
29C2×#14583809C2211yescharacteristic, complemented
30#19149668C2211yescharacteristic, complemented
31#19149668C2211yescharacteristic, complemented
32#19149664—21no—
33#19149662—21no—
34#19149655C2211yescharacteristic, complemented
35#19149654—21no—
36#19149654C2211yescharacteristic
37#19149653C2211yescharacteristic
38#19149651—21no—
39#19149649—21no—
40#19149647—21no—
41D4×Dic173C2211yescharacteristic, complemented
42#19149641C2211yescharacteristic, complemented
43C4×C22⋊D173C2211yescharacteristic
44C2×#14583803—21no—
45C22⋊C4×D173—21no—
46#19149612C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic, complemented
51#19149599C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53C8×Dic173C2211yescharacteristic, complemented
54C2×#41×C173C2211yescharacteristic
55C2×Q16×C173C2211yescharacteristic, complemented
56C2×SD16×C173C2211yescharacteristic, complemented
57C22⋊Q8×C173C2211yescharacteristic, complemented
58#103×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic, complemented
60C22⋊C8×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63#14583827—22no—
64#14583827—41no—
65C2×D692—22no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—41no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173D411yescharacteristic
76D4×D173—22no—
77D4×D173—41no—
78C22×Dic173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173C2311yescharacteristic
82#14583813—22no—
83#14583813—22no—
84#14583813—22no—
85D1384—22no—
86D1384—41no—
87#14583811—22no—
88#14583811D414yescomplemented
89#14583810—22no—
90#14583809—22no—
91C2×#13063436C2311yescharacteristic
92C2×#13063436C2311yescharacteristic
93C4⋊Dic173C2311yescharacteristic
94#14583805—21no—
95#14583805D411yescharacteristic
96C22⋊Dic173—21no—
97C22⋊Dic173D411yescharacteristic
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107C4×Dic173C2311yescharacteristic
108#41×C173—22no—
109#41×C173—41no—
110C2×Q8×C173C2311yescharacteristic
111C2×Q8×C173C2311yescharacteristic
112C2×D4×C173—21no—
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—41no—
117SD16×C173—22no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173—21no—
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—81no—
131C22×D173—81no—
132C22×D173C2×D411yescharacteristic
133D692—21no—
134D692—21no—
135D692—21no—
136D692—21no—
137D692—21no—
138D692—42no—
139D692—42no—
140D692—81no—
141D692C2×D412yes—
142C22⋊D173—41no—
143C22⋊D173—42no—
144C22⋊D173—81no—
145C2×Dic173—21no—
146C2×Dic173—81no—
147C2×Dic173C2×D411yescharacteristic
148C2×Dic173C2×D411yescharacteristic
149C2×Dic173C2×D411yescharacteristic
150C4×D173—21no—
151C4×D173—21no—
152C4×D173—42no—
153C4×D173—81no—
154#13063436—21no—
155#13063436C2×D412yes—
156C23×C173—21no—
157C23×C173C2×D411yescharacteristic
158Q8×C173—21no—
159Q8×C173—21no—
160Q8×C173—21no—
161Q8×C173—21no—
162D4×C173—21no—
163D4×C173—42no—
164D4×C173C2×D412yes—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—42no—
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173—21no—
175C22×C173—21no—
176C22×C173—22no—
177C22×C173—21no—
178C22×C173—21no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173C22×D411yescharacteristic
182C2×D173—21no—
183C2×D173—22no—
184C2×D173—21no—
185C2×D173—21no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—84no—
195C2×D173—84no—
196C4×C173—21no—
197C4×C173—41no—
198C4×C173—41no—
199C4×C173—41no—
200C4×C173C22×D411yescharacteristic
201C4×C173H(5,𝔽2)11yescharacteristic
202Dic173—21no—
203Dic173—21no—
204Dic173—21no—
205Dic173—41no—
206C2×C173—41no—
207C2×C173—42no—
208C2×C173D4211yescharacteristic
209C2×C173#57511yescharacteristic
210C2×C173#57611yescharacteristic
211D173—41no—
212D173—42no—
213D173—81no—
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#546—1731no—
221#495—1731no—
222#468—1731no—
223#468—1731no—
224#463—1731no—
225#463—1731no—
226#452—1731no—
227#452D17311yescharacteristic, complemented
228#447—1731no—
229#447—1731no—
230C4×D8—1731no—
231#436—1731no—
232C2×#41—1731no—
233C2×#41C2×D17311yescharacteristic
234C22×D4—1731no—
235C22×D4—1731no—
236AGL(1,ℤ/8)—3464no—
237AGL(1,ℤ/8)—3464no—
238C2×Q16C2×D17311yescharacteristic, complemented
239C2×SD16—1731no—
240C2×SD16—1731no—
241C2×SD16—1731no—
242C2×SD16C2×D17311yescharacteristic, complemented
243C2×D8—1731no—
244C2×D8—1731no—
245C2×D8—1731no—
246C2×D8—1731no—
247C2×M16—1731no—
248C2×M16—1731no—
249#124—1731no—
250#124—1731no—
251#123—3461no—
252#123—3461no—
253C22⋊Q8—1731no—
254C22⋊Q8C2×D17311yescharacteristic, complemented
255#121—3461no—
256#121—3461no—
257#121—1731no—
258#121—1731no—
259#120—3461no—
260#120—3461no—
261C4×D4—1731no—
262C4×D4—1731no—
263C4×D4—1731no—
264C2×C22⋊C4—3461no—
265C2×C22⋊C4—3461no—
266#107—1731no—
267C4⋊C8—1731no—
268#103—1731no—
269#103C2×D17311yescharacteristic, complemented
270#102—1731no—
271#102—1731no—
272#102—1731no—
273#102C2×D17311yescharacteristic, complemented
274C22⋊C8—1731no—
275C22⋊C8C2×D17311yescharacteristic, complemented
276C4×C8—1731no—
277C24—3461no—
278C24—3461no—
279#41—22no—
280#41—41no—
281#41—3462no—
282#41—6921no—
283C2×Q8C22×D17311yescharacteristic
284C2×Q8C22×D17311yescharacteristic
285C2×D4—21no—
286C2×D4—3462no—
287C2×D4—3461no—
288C2×D4—3461no—
289C2×D4—3462no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—1731no—
294C2×D4—1731no—
295C2×D4—1731no—
296C2×D4—1731no—
297C2×D4—6921no—
298C2×D4—6921no—
299C2×D4C22×D17311yescharacteristic
300C22×C4—21no—
301C22×C4—3461no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4—1731no—
305C22×C4C22×D17311yescharacteristic
306Q16—41no—
307SD16—22no—
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×C8C22×D17311yescharacteristic
320C2×C8C22×D17311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—3461no—
323C4⋊C4—1731no—
324C4⋊C4—1731no—
325C4⋊C4C22×D17311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—3461no—
335C22⋊C4—3461no—
336C22⋊C4—1731no—
337C22⋊C4—1731no—
338C42—1731no—
339C23—21no—
340C23—3461no—
341C23—3461no—
342C23—3461no—
343C23—13841no—
344C23—13841no—
345C23—1731no—
346C23—6922no—
347C23—6922no—
348C23—6922no—
349C23—6922no—
350C23D4×D17311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—3461no—
358D4—3461no—
359D4—3461no—
360D4—3461no—
361D4—3461no—
362D4—13841no—
363D4—13841no—
364D4—1732no—
365D4—6922no—
366D4—6922no—
367D4—6921no—
368D4—6922no—
369D4D4×D17312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—3461no—
377C2×C4—3461no—
378C2×C4—3461no—
379C2×C4—13841no—
380C2×C4—13841no—
381C2×C4—1731no—
382C2×C4—1731no—
383C2×C4—1731no—
384C2×C4—6922no—
385C2×C4D4×D17311yescharacteristic
386C2×C4C23×D17311yescharacteristic
387C8—21no—
388C8—21no—
389C8—3461no—
390C8—1732no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—3461no—
398C22—3461no—
399C22—3461no—
400C22—3462no—
401C22—13844no—
402C22—13844no—
403C22—6921no—
404C22—6921no—
405C22—6921no—
406C22—6921no—
407C22—6921no—
408C22—6921no—
409C22—6921no—
410C22—6921no—
411C22C2×D4×D17311yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—3461no—
417C4—3461no—
418C4—3461no—
419C4—6921no—
420C4C2×D4×D17311yescharacteristic
421C4#1914972411yescharacteristic
422C2—41no—
423C2—42no—
424C2—13841no—
425C2—13841no—
426C2—6921no—
427C2—6922no—
428C2#2601212311yescharacteristic
429C2#2601229011yescharacteristic
430C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
hypercenter#452D173
hypocenterC173#3273
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149704C2semidirect
3. #26012132C2semidirect
4. #26012129C2semidirect
5. #26011876C2semidirect
6. #26011740C2semidirect
7. #26011739C2semidirect
8. #26011709C2semidirect
9. #26011708C2semidirect
10. #26011652C2semidirect
11. #26011627C2semidirect
12. #26011626C2semidirect
13. #26011571C2semidirect
14. #26011566C2semidirect
15. #452×C173C2semidirect
17. C22×D692C22semidirect
18. C2×D4×D173C22semidirect
21. C2×#14583813C22semidirect
22. C2×#14583813C22semidirect
23. C2×#14583813C22semidirect
28. C2×#14583810C22semidirect
29. C2×#14583809C22semidirect
30. #19149668C22semidirect
31. #19149668C22semidirect
34. #19149655C22semidirect
41. D4×Dic173C22semidirect
42. #19149641C22semidirect
46. #19149612C22semidirect
47. #19149608C22semidirect
48. #19149603C22semidirect
49. #19149601C22semidirect
50. #19149601C22semidirect
51. #19149599C22semidirect
52. #19149591C22semidirect
53. C8×Dic173C22semidirect
55. C2×Q16×C173C22semidirect
56. C2×SD16×C173C22semidirect
57. C22⋊Q8×C173C22semidirect
58. #103×C173C22semidirect
59. #102×C173C22semidirect
60. C22⋊C8×C173C22semidirect
88. #14583811D4semidirect
215. C173#3273semidirect
227. #452D173semidirect
238. C2×Q16C2×D173semidirect
242. C2×SD16C2×D173semidirect
254. C22⋊Q8C2×D173semidirect
269. #103C2×D173semidirect
273. #102C2×D173semidirect
275. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×#191497042
#260121322
#260121292
#260118762
#260117402
#260117392
#260117092
#260117082
#260116522
#260116272
#260116262
#260115712
#260115662
#452×C1732
#3273173