Finite Group Databasebeta

← #36365085

Subgroups — #36365085

Order 22144.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149706C211yesmaximal, characteristic, complemented
2C2×#19149701C211yesmaximal, characteristic, complemented
3#26012143C211yesmaximal, characteristic, complemented
4#26012138C211yesmaximal, characteristic, complemented
5#26011882C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic, complemented
7#26011705C211yesmaximal, characteristic, complemented
8#26011695C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011643C211yesmaximal, characteristic, complemented
12#26011634C211yesmaximal, characteristic, complemented
13#26011602C211yesmaximal, characteristic, complemented
14#26011562C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583826C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic
20#19149706—24no—
21#19149701—24no—
22C2×Dic692C2211yescharacteristic, complemented
23C2×#14583816C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic
25C2×#14583815C2211yescharacteristic, complemented
26C2×#14583814C2211yescharacteristic
27C2×#14583813C2211yescharacteristic
28C2×D1384C2211yescharacteristic
29C2×#14583811C2211yescharacteristic
30C2×#14583809C2211yescharacteristic
31#19149677C2211yescharacteristic, complemented
32#19149666C2211yescharacteristic
33#19149664—21no—
34#19149661—21no—
35#19149661—21no—
36#19149659C2211yescharacteristic
37#19149659C2211yescharacteristic
38#19149652—21no—
39#19149650—21no—
40#19149647—21no—
41#19149641C2211yescharacteristic
42#19149641C2211yescharacteristic, complemented
43C4×C22⋊D173C2211yescharacteristic
44C4⋊C4×D173—21no—
45C22⋊C4×D173—21no—
46#19149608C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149599C2211yescharacteristic
50#19149599C2211yescharacteristic, complemented
51C22⋊#13063436C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic
53#19149589C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic
55C22×C8×C173C2211yescharacteristic, complemented
56C22⋊Q8×C173C2211yescharacteristic, complemented
57#121×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64#14583826—22no—
65#14583826—22no—
66#14583826—41no—
67#14583826—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70C2×D692—21no—
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173D411yescharacteristic
77D4×D173—22no—
78D4×D173—41no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173—21no—
84C2×C4×D173C2311yescharacteristic
85Dic692—22no—
86#14583816—41no—
87#14583815—22no—
88#14583815—22no—
89#14583814—41no—
90#14583813—22no—
91D1384D414yescomplemented
92#14583811—22no—
93#14583809—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C4⋊Dic173—21no—
97C4⋊Dic173—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803D411yescharacteristic
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114SD16×C173—22no—
115C2×C8×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173—21no—
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—21no—
133D692—21no—
134D692—21no—
135D692—42no—
136D692C2×D412yes—
137D692C2×D412yes—
138C22⋊D173—41no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—42no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C2×Dic173C2×D411yescharacteristic
147C4×D173—21no—
148C4×D173—21no—
149C4×D173—42no—
150C4×D173—42no—
151C4×D173—42no—
152C4×D173—42no—
153C4×D173—81no—
154#13063436—21no—
155#13063436—21no—
156C23×C173—21no—
157C23×C173C2×D411yescharacteristic
158Q8×C173—21no—
159Q8×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—81no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
180C2×D173—22no—
181C2×D173—22no—
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—41no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—84no—
189C4×C173—21no—
190C4×C173—41no—
191C4×C173—41no—
192C4×C173—41no—
193C4×C173C22×D411yescharacteristic
194C4×C173H(5,𝔽2)11yescharacteristic
195Dic173—21no—
196Dic173—21no—
197Dic173—21no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—41no—
201C2×C173—81no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—41no—
206D173—42no—
207D173—42no—
208C173#326711yescharacteristic, complemented, Sylow
209#3267—1731nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1731no—
211C2×#135—1731no—
212#547—1731no—
213#546—1731no—
214#497—1731no—
215#470—1731no—
216#465D17311yescharacteristic, complemented
217#463—1731no—
218#461—1731no—
219#452—1731no—
220#452—1731no—
221#450—1731no—
222#449—1731no—
223#442—1731no—
224#435—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C2×#41—1731no—
228C22×D4—1731no—
229AGL(1,ℤ/8)—3464no—
230#135—3464no—
231C2×Q16—1731no—
232C2×Q16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×SD16—1731no—
237C2×SD16C2×D17311yescharacteristic
238C2×D8—1731no—
239C2×D8—1731no—
240C2×M16—1731no—
241C22×C8C2×D17311yescharacteristic, complemented
242C4⋊Q8—1731no—
243#124—1731no—
244#123—3461no—
245#123—3461no—
246#123—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8—1731no—
249C22⋊Q8C2×D17311yescharacteristic, complemented
250#121—3461no—
251#121—3461no—
252#121C2×D17311yescharacteristic
253#120—3461no—
254C4×D4—1731no—
255C4×D4—1731no—
256C4×D4—1731no—
257C2×C4⋊C4—3461no—
258C2×C22⋊C4—3461no—
259#106C2×D17311yescharacteristic, complemented
260C4⋊C8—1731no—
261#103—1731no—
262#103—1731no—
263#103C2×D17311yescharacteristic, complemented
264#102—1731no—
265#102—1731no—
266#102C2×D17311yescharacteristic
267C22⋊C8—1731no—
268C22⋊C8—1731no—
269#97—1731no—
270C24—3461no—
271#41—3462no—
272#41—3462no—
273#41—3462no—
274#41—6921no—
275#41—6921no—
276#41—6921no—
277C2×Q8—1731no—
278C2×Q8—1731no—
279C2×Q8C22×D17311yescharacteristic
280C2×D4—21no—
281C2×D4—3461no—
282C2×D4—3462no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—1731no—
287C2×D4—1731no—
288C2×D4—1731no—
289C2×D4—6921no—
290C2×D4C22×D17311yescharacteristic
291C22×C4—3461no—
292C22×C4—3461no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4C22×D17311yescharacteristic
298Q16—3462no—
299Q16—6921no—
300SD16—22no—
301SD16—3462no—
302SD16—3462no—
303SD16—3462no—
304SD16—6921no—
305D8—3462no—
306D8—1734no—
307M16—3462no—
308C2×C8—22no—
309C2×C8—1731no—
310C2×C8—1731no—
311C2×C8C22×D17311yescharacteristic
312C2×C8C22×D17311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—3461no—
317C4⋊C4—3461no—
318C4⋊C4—1731no—
319C4⋊C4C22×D17311yescharacteristic
320C4⋊C4C22×D17311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—1731no—
330C22⋊C4—1731no—
331C42—1731no—
332C23—21no—
333C23—3461no—
334C23—3461no—
335C23—13841no—
336C23—1731no—
337C23—6922no—
338C23—6922no—
339C23D4×D17311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3461no—
343Q8—3461no—
344Q8—3461no—
345Q8—3461no—
346D4—21no—
347D4—21no—
348D4—81no—
349D4—3461no—
350D4—3461no—
351D4—1732no—
352D4—1732no—
353D4—6922no—
354D4—6921no—
355D4—6922no—
356D4—6922no—
357D4—6922no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—13841no—
368C2×C4—1731no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—6922no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4—6922no—
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—3461no—
379C8—3461no—
380C8D4×D17312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3461no—
388C22—3461no—
389C22—3461no—
390C22—3462no—
391C22—3462no—
392C22—13844no—
393C22—6921no—
394C22—6921no—
395C22—6921no—
396C22—6921no—
397C22C2×D4×D17311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—3461no—
403C4—3461no—
404C4—3461no—
405C4—6921no—
406C4—6921no—
407C4C2×D4×D17311yescharacteristic
408C4#1914972511yescharacteristic
409C2—41no—
410C2—81no—
411C2—6921no—
412C2—6922no—
413C2—6922no—
414C2#2601212511yescharacteristic
415C2#2601228811yescharacteristic
416C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
hypercenter#465D173
hypocenterC173#3267
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149701C2semidirect
3. #26012143C2semidirect
4. #26012138C2semidirect
5. #26011882C2semidirect
6. #26011750C2semidirect
7. #26011705C2semidirect
8. #26011695C2semidirect
9. #26011656C2semidirect
10. #26011651C2semidirect
11. #26011643C2semidirect
12. #26011634C2semidirect
13. #26011602C2semidirect
14. #26011562C2semidirect
15. #465×C173C2semidirect
18. C2×#14583826C22semidirect
22. C2×Dic692C22semidirect
23. C2×#14583816C22semidirect
25. C2×#14583815C22semidirect
31. #19149677C22semidirect
42. #19149641C22semidirect
46. #19149608C22semidirect
47. #19149602C22semidirect
48. #19149602C22semidirect
50. #19149599C22semidirect
51. C22⋊#13063436C22semidirect
53. #19149589C22semidirect
55. C22×C8×C173C22semidirect
56. C22⋊Q8×C173C22semidirect
58. #106×C173C22semidirect
59. #103×C173C22semidirect
91. D1384D4semidirect
208. C173#3267semidirect
216. #465D173semidirect
241. C22×C8C2×D173semidirect
249. C22⋊Q8C2×D173semidirect
259. #106C2×D173semidirect
263. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×#191497012
#260121432
#260121382
#260118822
#260117502
#260117052
#260116952
#260116562
#260116512
#260116432
#260116342
#260116022
#260115622
#465×C1732
#3267173