Finite Group Databasebeta

← #36363240

Subgroups — #36363240

Order 22144.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic, complemented
2C2×#19149687C211yesmaximal, characteristic, complemented
3#26012070C211yesmaximal, characteristic, complemented
4#26012057C211yesmaximal, characteristic, complemented
5#26011743C211yesmaximal, characteristic, complemented
6#26011737C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011725C211yesmaximal, characteristic, complemented
9#26011724C211yesmaximal, characteristic, complemented
10#26011721C211yesmaximal, characteristic, complemented
11#26011718C211yesmaximal, characteristic, complemented
12#26011530C211yesmaximal, characteristic, complemented
13#26011484C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#469×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149703—24no—
20C2×#14583815C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23#19149687—24no—
24C2×#14583809C2211yescharacteristic
25C2×#14583809C2211yescharacteristic, complemented
26C22×#13063436C2211yescharacteristic, complemented
27C2×C8×D173C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic, complemented
29#19149675—21no—
30#19149665—21no—
31#19149659C2211yescharacteristic, complemented
32#19149658C2211yescharacteristic, complemented
33#19149656—21no—
34#19149654C2211yescharacteristic
35#19149653C2211yescharacteristic
36#19149652—21no—
37#19149651—21no—
38#19149650—21no—
39#19149650C2211yescharacteristic
40#19149650C2211yescharacteristic, complemented
41#19149649—21no—
42#19149647—21no—
43#19149637C2211yescharacteristic
44C4⋊C4×D173C2211yescharacteristic
45#19149613C2211yescharacteristic, complemented
46#19149612C2211yescharacteristic, complemented
47#19149610C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149602C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic
51#19149600C2211yescharacteristic
52#19149600C2211yescharacteristic, complemented
53#19149599C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55C2×M16×C173C2211yescharacteristic, complemented
56#121×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic, complemented
58#106×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583826—22no—
63#14583826—22no—
64#14583825—22no—
65#14583825—22no—
66C2×Dic346D411yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×D692—21no—
69C2×D692D411yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173D411yescharacteristic
77C2×C22⋊D173D411yescharacteristic
78D4×D173—22no—
79D4×D173—22no—
80C22×Dic173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83C2×C4×D173C2311yescharacteristic
84#14583815—22no—
85#14583814—22no—
86#14583811—22no—
87#14583809—22no—
88#14583809D414yescomplemented
89C2×#13063436—22no—
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C8×D173—22no—
93C8×D173—22no—
94C4⋊Dic173—21no—
95C4⋊Dic173C2311yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97#14583805—21no—
98#14583805—21no—
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106C4×Dic173—21no—
107C2×D4×C173—21no—
108C2×D4×C173—21no—
109C2×D4×C173C2311yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112D8×C173—22no—
113M16×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173C2311yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22×D173—21no—
121C22×D173—42no—
122C22×D173—81no—
123C22×D173—81no—
124C22×D173C2×D411yescharacteristic
125C22×D173C2×D411yescharacteristic
126Dic346—22no—
127Dic346—21no—
128Dic346—21no—
129D692—22no—
130D692—21no—
131D692—21no—
132D692—81no—
133C22⋊D173—22no—
134C22⋊D173—22no—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C22⋊D173—81no—
138C22⋊D173—81no—
139C22⋊D173—81no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—42no—
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—81no—
149C4×D173C2×D412yes—
150C4×D173#4112yes—
151#13063436—21no—
152#13063436C2×D412yes—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173C2×D411yescharacteristic
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—81no—
161D4×C173—81no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173C2×D411yescharacteristic
167C2×C4×C173C2411yescharacteristic
168C8×C173—21no—
169C8×C173C2×D412yes—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173C22×D411yescharacteristic
180C2×D173—21no—
181C2×D173—21no—
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—84no—
189C2×D173#12112yes—
190C4×C173—21no—
191C4×C173—41no—
192C4×C173—41no—
193C4×C173C22×D411yescharacteristic
194C4×C173C2×#4111yescharacteristic
195Dic173—21no—
196Dic173—41no—
197Dic173—41no—
198Dic173—41no—
199Dic173#12112yes—
200C2×C173—41no—
201C2×C173—81no—
202C2×C173—81no—
203C2×C173C2×#12111yescharacteristic
204C2×C173#57611yescharacteristic
205C2×C173#57711yescharacteristic
206D173—22no—
207D173—41no—
208D173—81no—
209C173#306211yescharacteristic, complemented, Sylow
210#3062—1731nomaximal, Sylow
211C2×AGL(1,ℤ/8)—1731no—
212C2×#131—1731no—
213#537—1731no—
214#535—1731no—
215#469—1731no—
216#469D17311yescharacteristic, complemented
217#468—1731no—
218#468—1731no—
219#466—1731no—
220#466—1731no—
221#465—1731no—
222#465—1731no—
223#428—1731no—
224#416—1731no—
225C2×#102—1731no—
226C2×#41—1731no—
227C2×#41—1731no—
228C22×D4—1731no—
229AGL(1,ℤ/8)—3464no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×D8—1731no—
233C2×D8C2×D17311yescharacteristic
234#131—3464no—
235C2×M16—1731no—
236C2×M16—1731no—
237C2×M16C2×D17311yescharacteristic, complemented
238C22×C8—1731no—
239C22×C8—1731no—
240C22×C8—1731no—
241#127—3461no—
242#124—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8—1731no—
246#121—3461no—
247#121—3461no—
248#121—3461no—
249#121—1731no—
250#121—1731no—
251#121—1731no—
252#121—1731no—
253#121C2×D17311yescharacteristic
254#121C2×D17311yescharacteristic, complemented
255#120—3461no—
256#120—3461no—
257#117—1731no—
258C2×C4⋊C4—1731no—
259#107—1731no—
260#107—1731no—
261#106—1731no—
262#106C2×D17311yescharacteristic, complemented
263#103—1731no—
264#103—1731no—
265#102—1731no—
266#102—1731no—
267#102—1731no—
268#102—1731no—
269#102C2×D17311yescharacteristic
270#102C2×D17311yescharacteristic, complemented
271C24—3461no—
272#41—3462no—
273#41—3462no—
274#41—3462no—
275#41—3462no—
276C2×Q8—1731no—
277C2×Q8—1731no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3462no—
286C2×D4—3462no—
287C2×D4—3461no—
288C2×D4—1731no—
289C2×D4—1731no—
290C2×D4—1731no—
291C2×D4—1731no—
292C2×D4C22×D17311yescharacteristic
293C2×D4C22×D17311yescharacteristic
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4—1731no—
298C22×C4C22×D17311yescharacteristic
299SD16—3462no—
300SD16—3462no—
301D8—22no—
302D8—3462no—
303M16—22no—
304M16—3462no—
305M16—1734no—
306C2×C8—3462no—
307C2×C8—3462no—
308C2×C8—3462no—
309C2×C8—1731no—
310C2×C8—1731no—
311C2×C8C22×D17311yescharacteristic
312C2×C8C22×D17311yescharacteristic
313C4⋊C4—3461no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—1731no—
317C4⋊C4—1731no—
318C4⋊C4C22×D17311yescharacteristic
319C4⋊C4C22×D17311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C42—3461no—
330C23—21no—
331C23—21no—
332C23—3461no—
333C23—13841no—
334C23—13841no—
335C23—1731no—
336C23—1731no—
337C23—6922no—
338C23D4×D17311yescharacteristic
339Q8—3462no—
340Q8—3461no—
341Q8—3461no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—3462no—
349D4—3462no—
350D4—3462no—
351D4—3461no—
352D4—3461no—
353D4—13841no—
354D4—13841no—
355D4—13841no—
356D4—13841no—
357D4—6922no—
358D4—6922no—
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—3461no—
368C2×C4—13841no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—1732no—
372C2×C4—1732no—
373C2×C4—6922no—
374C2×C4D4×D17311yescharacteristic
375C2×C4C23×D17311yescharacteristic
376C8—21no—
377C8—3461no—
378C8—1732no—
379C8D4×D17312yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3461no—
390C22—3461no—
391C22—3461no—
392C22—3461no—
393C22—3461no—
394C22—13844no—
395C22—1732no—
396C22—6921no—
397C22—6921no—
398C22—6921no—
399C22C2×D4×D17311yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—3461no—
404C4—1732no—
405C4—6921no—
406C4—6921no—
407C4—6921no—
408C4C2×D4×D17311yescharacteristic
409C4#41×D17311yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—3462no—
414C2—13841no—
415C2—6921no—
416C2#121×D17311yescharacteristic
417C2#2601229511yescharacteristic
418C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3062
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×#19149687C2semidirect
3. #26012070C2semidirect
4. #26012057C2semidirect
5. #26011743C2semidirect
6. #26011737C2semidirect
7. #26011736C2semidirect
8. #26011725C2semidirect
9. #26011724C2semidirect
10. #26011721C2semidirect
11. #26011718C2semidirect
12. #26011530C2semidirect
13. #26011484C2semidirect
14. #102×D173C2semidirect
15. #469×C173C2semidirect
25. C2×#14583809C22semidirect
26. C22×#13063436C22semidirect
28. C2×C8×D173C22semidirect
31. #19149659C22semidirect
32. #19149658C22semidirect
40. #19149650C22semidirect
45. #19149613C22semidirect
46. #19149612C22semidirect
47. #19149610C22semidirect
48. #19149604C22semidirect
49. #19149602C22semidirect
52. #19149600C22semidirect
55. C2×M16×C173C22semidirect
57. #121×C173C22semidirect
58. #106×C173C22semidirect
60. #102×C173C22semidirect
88. #14583809D4semidirect
209. C173#3062semidirect
216. #469D173semidirect
237. C2×M16C2×D173semidirect
254. #121C2×D173semidirect
262. #106C2×D173semidirect
270. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×#191496872
#260120702
#260120572
#260117432
#260117372
#260117362
#260117252
#260117242
#260117212
#260117182
#260115302
#260114842
#102×D1732
#469×C1732
#3062173