Finite Group Databasebeta

← #36365578

Subgroups — #36365578

Order 22144.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D692C211yesmaximal, characteristic
2#26011857C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic, complemented
4#26011804C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011739C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011691C211yesmaximal, characteristic, complemented
9#26011641C211yesmaximal, characteristic, complemented
10#26011641C211yesmaximal, characteristic, complemented
11#26011589C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011495C211yesmaximal, characteristic, complemented
14C2×#19149599C211yesmaximal, characteristic, complemented
15#552×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×D4×D173—21no—
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic, complemented
21C2×#14583813C2211yescharacteristic, complemented
22C2×#14583810C2211yescharacteristic, complemented
23C22×#13063436C2211yescharacteristic, complemented
24#19149676C2211yescharacteristic
25#19149654—21no—
26#19149653C2211yescharacteristic
27#19149653C2211yescharacteristic
28#19149649—21no—
29#19149649—21no—
30C4×D692C2211yescharacteristic
31#19149612C2211yescharacteristic, complemented
32#19149612C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149603C2211yescharacteristic, complemented
36#19149599C2211yescharacteristic
37#19149599C2211yescharacteristic
38#19149599C2211yescharacteristic
39#19149599C2211yescharacteristic, complemented
40#19149599C2211yescharacteristic, complemented
41#19149599C2211yescharacteristic, complemented
42#19149599C2211yescharacteristic, complemented
43#19149599C2211yescharacteristic, complemented
44#19149599C2211yescharacteristic, complemented
45C22⋊#13063436C2211yescharacteristic, complemented
46C22⋊#13063436C2211yescharacteristic, complemented
47#19149590C2211yescharacteristic, complemented
48#126×C173—21no—
49#125×C173—21no—
50#125×C173C2211yescharacteristic, complemented
51#123×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic, complemented
53C22⋊Q8×C173C2211yescharacteristic, complemented
54C4×Q8×C173C2211yescharacteristic, complemented
55C4×D4×C173C2211yescharacteristic
56#117×C173—21no—
57#117×C173C2211yescharacteristic, complemented
58C2×C4⋊C4×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173—21no—
61C2×D692—22no—
62C2×D692—22no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—41no—
68C2×D692—41no—
69C2×D692C2311yescharacteristic
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74D4×D173—44no—
75C2×C4×D173—21no—
76#14583813—22no—
77#14583813—41no—
78#14583813—41no—
79#14583810—41no—
80C2×#13063436—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89C2×Q8×C173C2311yescharacteristic
90C2×D4×C173D411yescharacteristic
91C22×C4×C173—21no—
92C22×C4×C173C2311yescharacteristic
93C22×C4×C173C2311yescharacteristic
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173—21no—
97C4⋊C4×C173—21no—
98C4⋊C4×C173—21no—
99C4⋊C4×C173—21no—
100C4⋊C4×C173D411yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C42×C173—21no—
110C42×C173—21no—
111C42×C173C2311yescharacteristic
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—42no—
117C22×D173—42no—
118C22×D173—42no—
119C22×D173—81no—
120C22×D173—81no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—42no—
127D692—42no—
128D692—42no—
129D692—81no—
130D692#4112yes—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C2×Dic173—21no—
134C4×D173—42no—
135#13063436—21no—
136#13063436—21no—
137#13063436—21no—
138#13063436—21no—
139C23×C173C2×D411yescharacteristic
140C23×C173C2×D411yescharacteristic
141Q8×C173—21no—
142Q8×C173—21no—
143D4×C173—22no—
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—21no—
151C2×C4×C173—21no—
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×C173AGL(1,ℤ/8)12yes—
164C22×C173C22×D411yescharacteristic
165C2×D173—22no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—84no—
175C2×D173—84no—
176C4×C173—21no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173—41no—
184C4×C173C2×#4111yescharacteristic
185C4×C173H(5,𝔽2)11yescharacteristic
186Dic173—41no—
187C2×C173—22no—
188C2×C173—41no—
189C2×C173#54611yescharacteristic
190C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C173#57711yescharacteristic
192D173—42no—
193D173—81no—
194D173—81no—
195C173#331011yescharacteristic, complemented, Sylow
196#3310—1731nomaximal, Sylow
197#552D17311yescharacteristic, complemented
198D42—1731no—
199#490—1731no—
200#482—1731no—
201#480—1731no—
202#478—1731no—
203#468—1731no—
204#465—1731no—
205#460—1731no—
206#450—1731no—
207#450—1731no—
208#440—1731no—
209#435—1731no—
210#418—1731no—
211C2×#102—1731no—
212C22×D4—3461no—
213C22×D4—1731no—
214C22×D4—1731no—
215C2×SD16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×M16—1731no—
219C22×C8—1731no—
220#127—1731no—
221#126—21no—
222#125—21no—
223#125C2×D17311yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D17311yescharacteristic, complemented
226C22⋊Q8C2×D17311yescharacteristic, complemented
227#121—3461no—
228#121—1731no—
229#121—1731no—
230#120—3461no—
231#120—3461no—
232C4×Q8C2×D17311yescharacteristic, complemented
233C4×D4—1731no—
234C4×D4C2×D17311yescharacteristic
235#117—21no—
236#117C2×D17311yescharacteristic, complemented
237C2×C4⋊C4C2×D17311yescharacteristic, complemented
238#107—1731no—
239#107—1731no—
240#106—1731no—
241C4⋊C8—1731no—
242#103—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97—1731no—
255C24—3461no—
256C24—3461no—
257C2×Q8C22×D17311yescharacteristic
258C2×D4—3462no—
259C2×D4—3462no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—1731no—
267C2×D4—1731no—
268C2×D4—1731no—
269C2×D4—6921no—
270C2×D4—6921no—
271C2×D4—6924no—
272C2×D4C22⋊D17311yescharacteristic
273C22×C4—21no—
274C22×C4—3461no—
275C22×C4C22×D17311yescharacteristic
276C22×C4C22×D17311yescharacteristic
277SD16—3462no—
278SD16—6921no—
279SD16—6921no—
280M16—6921no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8—6921no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1731no—
293C4⋊C4C22⋊D17311yescharacteristic
294C4⋊C4C22×D17311yescharacteristic
295C4⋊C4C22×D17311yescharacteristic
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4C22⋊D17311yescharacteristic
304C22⋊C4C22⋊D17311yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D17311yescharacteristic
308C23—3461no—
309C23—3461no—
310C23—3461no—
311C23—13841no—
312C23—13841no—
313C23—6922no—
314C23—6922no—
315C23—6922no—
316C23—6922no—
317C23C2×C22⋊D17311yescharacteristic
318C23C2×C22⋊D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—3461no—
323D4—3461no—
324D4—3461no—
325D4—3461no—
326D4—3461no—
327D4—13841no—
328D4—1732no—
329D4—6922no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3461no—
344C2×C4—6922no—
345C2×C4C2×C22⋊D17311yescharacteristic
346C2×C4C2×C22⋊D17311yescharacteristic
347C2×C4C2×C22⋊D17311yescharacteristic
348C2×C4C2×C22⋊D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—3461no—
351C8—3461no—
352C8—3461no—
353C8—3461no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—3462no—
360C22—13844no—
361C22—13844no—
362C22—6921no—
363C22—6921no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22#1914970412yes—
371C22C22×C22⋊D17311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—6921no—
381C4#41×D17311yescharacteristic
382C4#1914972511yescharacteristic
383C2—22no—
384C2—41no—
385C2—13841no—
386C2—13841no—
387C2—6922no—
388C2#2601212911yescharacteristic
389C2C2×#1914970411yescharacteristic
390C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#552×C173C2
socleC22×C173C22×D4
prime radicalD4×D692C2
hypercenter#552D173
hypocenterC173#3310
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #26011857C2semidirect
3. #26011815C2semidirect
4. #26011804C2semidirect
5. #26011788C2semidirect
6. #26011739C2semidirect
7. #26011722C2semidirect
8. #26011691C2semidirect
9. #26011641C2semidirect
10. #26011641C2semidirect
11. #26011589C2semidirect
12. #26011563C2semidirect
13. #26011495C2semidirect
14. C2×#19149599C2semidirect
15. #552×C173C2semidirect
20. C2×#14583813C22semidirect
21. C2×#14583813C22semidirect
22. C2×#14583810C22semidirect
23. C22×#13063436C22semidirect
31. #19149612C22semidirect
32. #19149612C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149603C22semidirect
39. #19149599C22semidirect
40. #19149599C22semidirect
41. #19149599C22semidirect
42. #19149599C22semidirect
43. #19149599C22semidirect
44. #19149599C22semidirect
45. C22⋊#13063436C22semidirect
46. C22⋊#13063436C22semidirect
47. #19149590C22semidirect
50. #125×C173C22semidirect
52. C22⋊Q8×C173C22semidirect
53. C22⋊Q8×C173C22semidirect
54. C4×Q8×C173C22semidirect
57. #117×C173C22semidirect
58. C2×C4⋊C4×C173C22semidirect
195. C173#3310semidirect
197. #552D173semidirect
223. #125C2×D173semidirect
225. C22⋊Q8C2×D173semidirect
226. C22⋊Q8C2×D173semidirect
232. C4×Q8C2×D173semidirect
236. #117C2×D173semidirect
237. C2×C4⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
D4×D6922
#260118572
#260118152
#260118042
#260117882
#260117392
#260117222
#260116912
#260116412
#260115892
#260115632
#260114952
C2×#191495992
#552×C1732
#3310173