Finite Group Databasebeta

← #36365063

Subgroups — #36365063

Order 22144.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149707C211yesmaximal, characteristic, complemented
2C22×#14583815C211yesmaximal, characteristic, complemented
3#26012150C211yesmaximal, characteristic, complemented
4D4×D692C211yesmaximal, characteristic, complemented
5#26011880C211yesmaximal, characteristic, complemented
6#26011737C211yesmaximal, characteristic, complemented
7#26011727C211yesmaximal, characteristic, complemented
8#26011707C211yesmaximal, characteristic, complemented
9#26011692C211yesmaximal, characteristic, complemented
10#26011644C211yesmaximal, characteristic, complemented
11#26011638C211yesmaximal, characteristic, complemented
12#26011632C211yesmaximal, characteristic, complemented
13#26011627C211yesmaximal, characteristic, complemented
14#26011593C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C22×Dic346C2211yescharacteristic, complemented
18C22×D692C2211yescharacteristic
19C22×D692C2211yescharacteristic
20C2×D4×D173—21no—
21#19149707—24no—
22C2×#14583815—24no—
23C2×#14583815C2211yescharacteristic
24C2×#14583815C2211yescharacteristic
25C2×#14583815C2211yescharacteristic, complemented
26C2×#14583815C2211yescharacteristic, complemented
27C2×#14583815C2211yescharacteristic, complemented
28C2×#14583815C2211yescharacteristic, complemented
29C2×D1384C2211yescharacteristic
30C2×D1384C2211yescharacteristic, complemented
31C2×D1384C2211yescharacteristic, complemented
32#19149676C2211yescharacteristic
33#19149667C2211yescharacteristic, complemented
34#19149665—21no—
35#19149658C2211yescharacteristic, complemented
36#19149657—21no—
37#19149654—21no—
38#19149653C2211yescharacteristic
39#19149653C2211yescharacteristic
40#19149653C2211yescharacteristic, complemented
41#19149651—21no—
42#19149649—21no—
43#19149649—21no—
44C4×Dic346C2211yescharacteristic, complemented
45C4×D692C2211yescharacteristic
46#19149639—21no—
47#19149613C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149604C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic, complemented
52#19149601C2211yescharacteristic, complemented
53#19149601C2211yescharacteristic, complemented
54C2×M16×C173C2211yescharacteristic, complemented
55C22×C8×C173C2211yescharacteristic, complemented
56C4×D4×C173C2211yescharacteristic
57C4⋊C8×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C22⋊C8×C173C2211yescharacteristic, complemented
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63#14583826—22no—
64#14583826—41no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70C2×D692—22no—
71C2×D692—22no—
72C2×D692—21no—
73C2×D692—21no—
74C2×D692—21no—
75C2×D692—21no—
76C2×D692—41no—
77C2×D692—41no—
78C2×D692C2311yescharacteristic
79C2×D692C2311yescharacteristic
80C2×D692C2311yescharacteristic
81C2×D692C2311yescharacteristic
82C2×C22⋊D173—21no—
83C2×C22⋊D173—21no—
84C2×C22⋊D173—21no—
85C2×C22⋊D173—21no—
86D4×D173—44no—
87C22×Dic173—21no—
88C2×C4×D173—21no—
89C2×C4×D173—21no—
90#14583815—22no—
91#14583815—22no—
92#14583815—22no—
93#14583815—22no—
94#14583815—41no—
95#14583815D414yescomplemented
96D1384—22no—
97D1384—22no—
98D1384—41no—
99C4⋊Dic173C2311yescharacteristic
100C4⋊Dic173C2311yescharacteristic
101#14583805—21no—
102#14583805—21no—
103C22⋊Dic173—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109#14583803—21no—
110C4×Dic173—21no—
111C2×D4×C173D411yescharacteristic
112C22×C4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114M16×C173—22no—
115C2×C8×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C22⋊C4×C173D411yescharacteristic
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—21no—
127C22×D173—21no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—42no—
131C22×D173—42no—
132C22×D173—81no—
133C22×D173—81no—
134Dic346—21no—
135Dic346—21no—
136Dic346—21no—
137Dic346—42no—
138Dic346C2×D412yescomplemented
139D692—21no—
140D692—21no—
141D692—21no—
142D692—21no—
143D692—21no—
144D692—21no—
145D692—21no—
146D692—42no—
147D692—42no—
148D692—42no—
149D692—81no—
150D692C2×D412yes—
151C22⋊D173—42no—
152C22⋊D173—42no—
153C22⋊D173—42no—
154C22⋊D173—42no—
155C2×Dic173—21no—
156C2×Dic173—21no—
157C2×Dic173—21no—
158C2×Dic173—21no—
159C2×Dic173—42no—
160C4×D173—42no—
161C4×D173—42no—
162C23×C173C2×D411yescharacteristic
163C23×C173C2×D411yescharacteristic
164D4×C173—22no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173C2×D411yescharacteristic
170C2×C4×C173C2×D411yescharacteristic
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2×D411yescharacteristic
173C2×C4×C173C2411yescharacteristic
174C8×C173—21no—
175C8×C173—21no—
176C8×C173—21no—
177C8×C173C2×D412yescomplemented
178C22×C173—21no—
179C22×C173—21no—
180C22×C173—21no—
181C22×C173—21no—
182C22×C173—21no—
183C22×C173AGL(1,ℤ/8)12yescomplemented
184C22×C173C22×D411yescharacteristic
185C2×D173—22no—
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—41no—
195C2×D173—41no—
196C2×D173—41no—
197C2×D173—84no—
198C2×D173—84no—
199C4×C173—21no—
200C4×C173—21no—
201C4×C173—21no—
202C4×C173—21no—
203C4×C173C22×D411yescharacteristic
204C4×C173H(5,𝔽2)11yescharacteristic
205Dic173—22no—
206Dic173—41no—
207Dic173—41no—
208Dic173—41no—
209C2×C173—22no—
210C2×C173—41no—
211C2×C173D4211yescharacteristic
212C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C173#57711yescharacteristic
214D173—42no—
215D173—81no—
216D173—81no—
217D173—81no—
218C173#326511yescharacteristic, complemented, Sylow
219#3265—1731nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1731no—
221C22×SD16—1731no—
222#548—1731no—
223D42—1731no—
224#496—1731no—
225#468—1731no—
226#466—1731no—
227#463—1731no—
228#460—1731no—
229#450—1731no—
230#449—1731no—
231#448—1731no—
232#447—1731no—
233#440—1731no—
234#435D17311yescharacteristic, complemented
235C2×#41—1731no—
236C22×Q8—1731no—
237C22×D4—3461no—
238C22×D4—1731no—
239C22×D4—1731no—
240AGL(1,ℤ/8)—3464no—
241C2×SD16—3464no—
242C2×SD16—1731no—
243C2×SD16—1731no—
244C2×SD16—1731no—
245C2×SD16—1731no—
246C2×SD16—1731no—
247C2×SD16—1731no—
248C2×D8—1731no—
249C2×D8—1731no—
250C2×D8—1731no—
251C2×M16C2×D17311yescharacteristic, complemented
252C22×C8C2×D17311yescharacteristic, complemented
253#127—1731no—
254#124—3461no—
255#124—1731no—
256C22⋊Q8—3461no—
257C22⋊Q8—1731no—
258#121—3461no—
259#121—3461no—
260#121—1731no—
261#121—1731no—
262#121—1731no—
263#120—3461no—
264#120—3461no—
265C4×Q8—1731no—
266C4×D4—3461no—
267C4×D4—1731no—
268C4×D4C2×D17311yescharacteristic
269#107—1731no—
270C4⋊C8C2×D17311yescharacteristic, complemented
271#103—1731no—
272#103—1731no—
273#102—1731no—
274#102—1731no—
275#102—1731no—
276#102—1731no—
277C22⋊C8C2×D17311yescharacteristic, complemented
278C22⋊C8C2×D17311yescharacteristic, complemented
279#97C2×D17311yescharacteristic, complemented
280C24—3461no—
281C24—3461no—
282#41—3462no—
283#41—6921no—
284C2×Q8—3462no—
285C2×Q8—3461no—
286C2×Q8—1731no—
287C2×Q8—1731no—
288C2×Q8—6921no—
289C2×D4—3462no—
290C2×D4—3462no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—3461no—
294C2×D4—3461no—
295C2×D4—3461no—
296C2×D4—3461no—
297C2×D4—3461no—
298C2×D4—3461no—
299C2×D4—1731no—
300C2×D4—1731no—
301C2×D4—1731no—
302C2×D4—1731no—
303C2×D4—6921no—
304C2×D4—6921no—
305C2×D4—6924no—
306C2×D4D69211yescharacteristic
307C22×C4—3461no—
308C22×C4—3461no—
309C22×C4—3461no—
310C22×C4C22×D17311yescharacteristic
311C22×C4C22×D17311yescharacteristic
312SD16—3462no—
313SD16—3462no—
314SD16—3462no—
315SD16—3462no—
316SD16—1734no—
317SD16—6921no—
318D8—3462no—
319D8—3462no—
320D8—6921no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D17311yescharacteristic
324C2×C8C22×D17311yescharacteristic
325C2×C8C22×D17311yescharacteristic
326C2×C8C22×D17311yescharacteristic
327C4⋊C4—3461no—
328C4⋊C4—3461no—
329C4⋊C4—1731no—
330C4⋊C4—1731no—
331C4⋊C4D69211yescharacteristic
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—3461no—
335C22⋊C4—3461no—
336C22⋊C4—3461no—
337C22⋊C4—3461no—
338C22⋊C4—3461no—
339C22⋊C4D69211yescharacteristic
340C22⋊C4D69211yescharacteristic
341C42—3461no—
342C42C22×D17311yescharacteristic
343C23—3461no—
344C23—3461no—
345C23—3461no—
346C23—3461no—
347C23—13841no—
348C23—13841no—
349C23—6922no—
350C23—6922no—
351C23—6922no—
352C23—6922no—
353C23C2×D69211yescharacteristic
354C23C2×D69211yescharacteristic
355Q8—3461no—
356Q8—3461no—
357Q8—3461no—
358Q8—1732no—
359Q8—6922no—
360D4—22no—
361D4—3461no—
362D4—3461no—
363D4—3461no—
364D4—3461no—
365D4—3461no—
366D4—3461no—
367D4—3461no—
368D4—13841no—
369D4—1732no—
370D4—6922no—
371D4—6922no—
372D4—6922no—
373D4—6922no—
374D4—6922no—
375D4—6922no—
376D4—6922no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—6922no—
386C2×C4—6922no—
387C2×C4—6922no—
388C2×C4C2×D69211yescharacteristic
389C2×C4C2×D69211yescharacteristic
390C2×C4C2×D69211yescharacteristic
391C2×C4C2×D69211yescharacteristic
392C2×C4C23×D17311yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D17312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—3462no—
403C22—13844no—
404C22—13844no—
405C22—6921no—
406C22—6921no—
407C22—6921no—
408C22—6921no—
409C22—6921no—
410C22—6921no—
411C22—6921no—
412C22—6921no—
413C22—6921no—
414C22—6921no—
415C22—6921no—
416C22#1914970712yescomplemented
417C22C22×D69211yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—3462no—
423C4—6921no—
424C4—6921no—
425C4—6921no—
426C4C2×D4×D17311yescharacteristic
427C4#1914972511yescharacteristic
428C2—22no—
429C2—41no—
430C2—13841no—
431C2—13841no—
432C2—13841no—
433C2—6922no—
434C2D4×D69211yescharacteristic
435C2C2×#1914970711yescharacteristic
436C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
hypercenter#435D173
hypocenterC173#3265
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19149707C2semidirect
2. C22×#14583815C2semidirect
3. #26012150C2semidirect
4. D4×D692C2semidirect
5. #26011880C2semidirect
6. #26011737C2semidirect
7. #26011727C2semidirect
8. #26011707C2semidirect
9. #26011692C2semidirect
10. #26011644C2semidirect
11. #26011638C2semidirect
12. #26011632C2semidirect
13. #26011627C2semidirect
14. #26011593C2semidirect
15. #435×C173C2semidirect
16. C2×#14583826C22semidirect
17. C22×Dic346C22semidirect
25. C2×#14583815C22semidirect
26. C2×#14583815C22semidirect
27. C2×#14583815C22semidirect
28. C2×#14583815C22semidirect
30. C2×D1384C22semidirect
31. C2×D1384C22semidirect
33. #19149667C22semidirect
35. #19149658C22semidirect
40. #19149653C22semidirect
44. C4×Dic346C22semidirect
47. #19149613C22semidirect
48. #19149604C22semidirect
49. #19149604C22semidirect
51. #19149601C22semidirect
52. #19149601C22semidirect
53. #19149601C22semidirect
54. C2×M16×C173C22semidirect
55. C22×C8×C173C22semidirect
57. C4⋊C8×C173C22semidirect
58. C22⋊C8×C173C22semidirect
59. C22⋊C8×C173C22semidirect
60. #97×C173C22semidirect
95. #14583815D4semidirect
138. Dic346C2×D4semidirect
177. C8×C173C2×D4semidirect
183. C22×C173AGL(1,ℤ/8)semidirect
218. C173#3265semidirect
234. #435D173semidirect
251. C2×M16C2×D173semidirect
252. C22×C8C2×D173semidirect
270. C4⋊C8C2×D173semidirect
277. C22⋊C8C2×D173semidirect
278. C22⋊C8C2×D173semidirect
279. #97C2×D173semidirect
396. C8D4×D173semidirect
416. C22#19149707semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
C22×#145838152
#260121502
D4×D6922
#260118802
#260117372
#260117272
#260117072
#260116922
#260116442
#260116382
#260116322
#260116272
#260115932
#435×C1732
#3265173