Finite Group Databasebeta

← #36365518

Subgroups — #36365518

Order 22144.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012137C211yesmaximal, characteristic, complemented
2#26011853C211yesmaximal, characteristic, complemented
3#26011821C211yesmaximal, characteristic, complemented
4#26011809C211yesmaximal, characteristic, complemented
5#26011778C211yesmaximal, characteristic, complemented
6#26011742C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011706C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011624C211yesmaximal, characteristic, complemented
11#26011601C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011490C211yesmaximal, characteristic, complemented
14C2×#19149599C211yesmaximal, characteristic, complemented
15#547×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583814C2211yescharacteristic, complemented
19C2×#14583811C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583810C2211yescharacteristic, complemented
22C22×#13063436C2211yescharacteristic
23#19149667C2211yescharacteristic
24#19149664—21no—
25#19149658C2211yescharacteristic
26#19149653C2211yescharacteristic
27#19149652—21no—
28#19149647—21no—
29C4×D692C2211yescharacteristic
30C22⋊C4×D173—21no—
31#19149610C2211yescharacteristic, complemented
32#19149610C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149602C2211yescharacteristic, complemented
36#19149602C2211yescharacteristic, complemented
37#19149602C2211yescharacteristic, complemented
38#19149600C2211yescharacteristic, complemented
39#19149599C2211yescharacteristic
40#19149599C2211yescharacteristic
41#19149599C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic, complemented
47#19149590C2211yescharacteristic, complemented
48C2×#41×C173—21no—
49C4⋊Q8×C173C2211yescharacteristic, complemented
50#123×C173—21no—
51#123×C173—21no—
52C22⋊Q8×C173—21no—
53#121×C173C2211yescharacteristic
54#121×C173C2211yescharacteristic, complemented
55C4×D4×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic, complemented
57C2×C4⋊C4×C173C2211yescharacteristic
58C2×C4⋊C4×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—22no—
64C2×D692—21no—
65C2×D692—41no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72#14583814—41no—
73#14583811—22no—
74#14583811—41no—
75#14583810—41no—
76C2×#13063436—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#41×C173—44no—
89C2×Q8×C173—21no—
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173D411yescharacteristic
93C2×D4×C173C2311yescharacteristic
94C22×C4×C173—21no—
95C22×C4×C173—21no—
96C22×C4×C173—21no—
97C22×C4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173D411yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C42×C173C2311yescharacteristic
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173—81no—
117Dic346—21no—
118Dic346—21no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692—42no—
123D692#4112yescomplemented
124C22⋊D173—42no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C4×D173—42no—
129C4×D173—81no—
130#13063436—21no—
131#13063436—21no—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135C23×C173C2×D411yescharacteristic
136C23×C173C2×D411yescharacteristic
137Q8×C173—42no—
138D4×C173—22no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—42no—
142D4×C173—42no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—42no—
151C2×C4×C173—42no—
152C2×C4×C173—42no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173AGL(1,ℤ/8)12yes—
167C22×C173C22×D411yescharacteristic
168C2×D173—22no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—84no—
175C4×C173—21no—
176C4×C173—21no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173C2×#4111yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—41no—
185Dic173—41no—
186C2×C173—22no—
187C2×C173—41no—
188C2×C173—81no—
189C2×C173#54611yescharacteristic
190C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C173#57711yescharacteristic
192D173—42no—
193D173—81no—
194C173#330611yescharacteristic, complemented, Sylow
195#3306—1731nomaximal, Sylow
196#547D17311yescharacteristic, complemented
197#546—1731no—
198#489—1731no—
199#483—1731no—
200#481—1731no—
201#476—1731no—
202#469—1731no—
203#465—1731no—
204#463—1731no—
205#449—1731no—
206#447—1731no—
207#442—1731no—
208#435—1731no—
209#417—1731no—
210C2×#102—1731no—
211C2×#41—21no—
212C2×#41—1731no—
213C22×D4—1731no—
214C2×SD16—1731no—
215C2×D8—1731no—
216C2×D8—1731no—
217C2×M16—1731no—
218C22×C8—1731no—
219C4⋊Q8C2×D17311yescharacteristic, complemented
220#124—1731no—
221#123—21no—
222#123—21no—
223#123—3461no—
224C22⋊Q8—21no—
225C22⋊Q8—1731no—
226#121—3461no—
227#121—1731no—
228#121C2×D17311yescharacteristic
229#121C2×D17311yescharacteristic, complemented
230#120—3461no—
231C4×D4—1731no—
232C4×D4C2×D17311yescharacteristic
233C4×D4C2×D17311yescharacteristic, complemented
234C2×C4⋊C4C2×D17311yescharacteristic
235C2×C4⋊C4C2×D17311yescharacteristic, complemented
236C2×C22⋊C4—3461no—
237#106—1731no—
238#106—1731no—
239#106—1731no—
240C4⋊C8—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251C22⋊C8—1731no—
252C22⋊C8—1731no—
253#97—1731no—
254C24—3461no—
255#41—44no—
256#41—3462no—
257#41—6921no—
258C2×Q8—21no—
259C2×Q8—1731no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—3462no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—1731no—
267C2×D4—1731no—
268C2×D4—6921no—
269C2×D4C22⋊D17311yescharacteristic
270C2×D4C22×D17311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—3461no—
275C22×C4—3461no—
276C22×C4C22×D17311yescharacteristic
277C22×C4C22×D17311yescharacteristic
278SD16—6921no—
279D8—3462no—
280D8—6921no—
281M16—6921no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8—1731no—
286C2×C8—6921no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4C22⋊D17311yescharacteristic
293C4⋊C4C22×D17311yescharacteristic
294C4⋊C4C22×D17311yescharacteristic
295C4⋊C4C22×D17311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4C22⋊D17311yescharacteristic
305C22⋊C4C22⋊D17311yescharacteristic
306C42C22×D17311yescharacteristic
307C23—21no—
308C23—3461no—
309C23—3461no—
310C23—13841no—
311C23—6922no—
312C23—6922no—
313C23C2×C22⋊D17311yescharacteristic
314C23C2×C22⋊D17311yescharacteristic
315Q8—42no—
316Q8—3461no—
317Q8—3461no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—3461no—
324D4—3461no—
325D4—3461no—
326D4—13841no—
327D4—1732no—
328D4—6922no—
329D4—6922no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—13841no—
343C2×C4—6922no—
344C2×C4C2×C22⋊D17311yescharacteristic
345C2×C4C2×C22⋊D17311yescharacteristic
346C2×C4C2×C22⋊D17311yescharacteristic
347C2×C4C2×C22⋊D17311yescharacteristic
348C2×C4C23×D17311yescharacteristic
349C8—3461no—
350C8—3461no—
351C8—3461no—
352C8—3461no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3462no—
362C22—13844no—
363C22—6921no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22#1914970512yes—
369C22C22×C22⋊D17311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—6921no—
378C4—6921no—
379C4#41×D17311yescharacteristic
380C4#1914972511yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—13841no—
385C2—6922no—
386C2#2601212911yescharacteristic
387C2C2×#1914970511yescharacteristic
388C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#547×C173C2
socleC22×C173C22×D4
hypercenter#547D173
hypocenterC173#3306
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #26012137C2semidirect
2. #26011853C2semidirect
3. #26011821C2semidirect
4. #26011809C2semidirect
5. #26011778C2semidirect
6. #26011742C2semidirect
7. #26011718C2semidirect
8. #26011706C2semidirect
9. #26011634C2semidirect
10. #26011624C2semidirect
11. #26011601C2semidirect
12. #26011563C2semidirect
13. #26011490C2semidirect
14. C2×#19149599C2semidirect
15. #547×C173C2semidirect
18. C2×#14583814C22semidirect
21. C2×#14583810C22semidirect
31. #19149610C22semidirect
32. #19149610C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149602C22semidirect
36. #19149602C22semidirect
37. #19149602C22semidirect
38. #19149600C22semidirect
46. C22⋊#13063436C22semidirect
47. #19149590C22semidirect
49. C4⋊Q8×C173C22semidirect
54. #121×C173C22semidirect
56. C4×D4×C173C22semidirect
58. C2×C4⋊C4×C173C22semidirect
123. D692#41semidirect
194. C173#3306semidirect
196. #547D173semidirect
219. C4⋊Q8C2×D173semidirect
229. #121C2×D173semidirect
233. C4×D4C2×D173semidirect
235. C2×C4⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121372
#260118532
#260118212
#260118092
#260117782
#260117422
#260117182
#260117062
#260116342
#260116242
#260116012
#260115632
#260114902
C2×#191495992
#547×C1732
#3306173