Finite Group Databasebeta

← #36365090

Subgroups — #36365090

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×#19149703C211yesmaximal, characteristic, complemented
2C2×#19149698C211yesmaximal, characteristic, complemented
3#26012145C211yesmaximal, characteristic, complemented
4#26012136C211yesmaximal, characteristic, complemented
5#26011885C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011706C211yesmaximal, characteristic, complemented
9#26011696C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011654C211yesmaximal, characteristic, complemented
12#26011643C211yesmaximal, characteristic, complemented
13#26011637C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#442×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583825C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic
20#19149703—24no—
21#19149698—24no—
22C2×Dic692C2211yescharacteristic, complemented
23C2×Dic692C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic
25C2×#14583815C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583814C2211yescharacteristic
28C2×#14583814C2211yescharacteristic, complemented
29C2×#14583811C2211yescharacteristic
30C2×#14583809C2211yescharacteristic
31C2×C8×D173C2211yescharacteristic, complemented
32#19149679C2211yescharacteristic, complemented
33#19149667C2211yescharacteristic
34#19149663—21no—
35#19149663—21no—
36#19149663—21no—
37#19149659C2211yescharacteristic
38#19149659C2211yescharacteristic
39#19149659C2211yescharacteristic, complemented
40#19149654C2211yescharacteristic
41#19149653—21no—
42#19149653—21no—
43#19149649—21no—
44C4×D692C2211yescharacteristic
45C2×C4⋊Dic173—21no—
46C2×#14583803—21no—
47C4⋊#13063436C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149602C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149599C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic, complemented
57#106×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic, complemented
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583825—22no—
63#14583825—22no—
64#14583825—22no—
65#14583825—41no—
66#14583825—41no—
67#14583825—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70C2×Dic346C2311yescharacteristic
71C2×D692—21no—
72C2×D692D411yescharacteristic
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78D4×D173—22no—
79D4×D173—41no—
80C22×Dic173—21no—
81C22×Dic173—21no—
82C22×Dic173—21no—
83C2×C4×D173C2311yescharacteristic
84C2×C4×D173C2311yescharacteristic
85Dic692—22no—
86Dic692—41no—
87#14583815—22no—
88#14583815—41no—
89#14583814—22no—
90#14583814—22no—
91#14583814—22no—
92#14583811—22no—
93#14583809—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C8×D173—22no—
97C4⋊Dic173—21no—
98C4⋊Dic173—21no—
99C4⋊Dic173—21no—
100C4⋊Dic173—21no—
101C4⋊Dic173—21no—
102C4⋊Dic173D411yescharacteristic
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109#14583803D411yescharacteristic
110#14583803D411yescharacteristic
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114C22×C4×C173—21no—
115D8×C173D414yescomplemented
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173C2311yescharacteristic
119C4⋊C4×C173C2311yescharacteristic
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173—81no—
127C22×D173C2×D411yescharacteristic
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—21no—
133Dic346—21no—
134Dic346—21no—
135D692—21no—
136D692—21no—
137D692—41no—
138D692—81no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—42no—
142C22⋊D173—42no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—21no—
146C2×Dic173—42no—
147C2×Dic173—42no—
148C2×Dic173—42no—
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C4×D173—21no—
152C4×D173—21no—
153C4×D173—21no—
154C4×D173—21no—
155#13063436—21no—
156#13063436—21no—
157C23×C173—21no—
158C23×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173C2×D412yes—
162D4×C173C2×D412yes—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—42no—
166C2×C4×C173—81no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—22no—
173C22×C173—22no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
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—
189C4×C173—21no—
190C4×C173—21no—
191C4×C173—41no—
192C4×C173—41no—
193C4×C173C22×D411yescharacteristic
194C4×C173H(5,𝔽2)11yescharacteristic
195Dic173—21no—
196Dic173—21no—
197Dic173—41no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—42no—
201C2×C173—42no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—41no—
206D173—41no—
207D173—81no—
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#465—1731no—
217#463—1731no—
218#461—1731no—
219#452—1731no—
220#452—1731no—
221#450—1731no—
222#449—1731no—
223#442D17311yescharacteristic, complemented
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×SD16—1731no—
238C2×D8—1731no—
239C2×D8C2×D17311yescharacteristic
240C2×M16—1731no—
241C22×C8—1731no—
242C4⋊Q8—1731no—
243#124—1731no—
244#123—3461no—
245#123—3461no—
246#123—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8—1731no—
249C22⋊Q8—1731no—
250#121—3461no—
251#121—3461no—
252#121—1731no—
253#120—3461no—
254C4×D4—1731no—
255C4×D4C2×D17311yescharacteristic
256C4×D4C2×D17311yescharacteristic, complemented
257C2×C4⋊C4—3461no—
258C2×C22⋊C4—3461no—
259#106C2×D17311yescharacteristic, complemented
260C4⋊C8—1731no—
261#103—1731no—
262#103—1731no—
263#103—1731no—
264#102—1731no—
265#102C2×D17311yescharacteristic
266#102C2×D17311yescharacteristic, complemented
267C22⋊C8—1731no—
268C22⋊C8—1731no—
269#97C2×D17311yescharacteristic, complemented
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×Q8—1731no—
280C2×D4—3461no—
281C2×D4—3462no—
282C2×D4—3461no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—1731no—
287C2×D4—1731no—
288C2×D4—6921no—
289C2×D4C22×D17311yescharacteristic
290C2×D4C22×D17311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4—1731no—
298Q16—3462no—
299Q16—6921no—
300SD16—3462no—
301SD16—3462no—
302SD16—3462no—
303SD16—3462no—
304SD16—6921no—
305D8—3462no—
306D8D69214yescomplemented
307M16—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—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—
331C42C22×D17311yescharacteristic
332C23—21no—
333C23—21no—
334C23—3461no—
335C23—13841no—
336C23—1731no—
337C23—1731no—
338C23—6922no—
339C23—6922no—
340Q8—3461no—
341Q8—3461no—
342Q8—3461no—
343Q8—3461no—
344Q8—3461no—
345Q8—3461no—
346D4—21no—
347D4—21no—
348D4—3461no—
349D4—3461no—
350D4—13841no—
351D4—6921no—
352D4—6922no—
353D4—6922no—
354D4—6922no—
355D4—6922no—
356D4C2×D69212yes—
357D4C2×D69212yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—6922no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4D4×D17311yescharacteristic
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—3461no—
379C8—3461no—
380C8C2×D69212yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3461no—
388C22—3461no—
389C22—3461no—
390C22—3461no—
391C22—3461no—
392C22—3461no—
393C22—13844no—
394C22—6921no—
395C22—6921no—
396C22—6921no—
397C22C2×D4×D17311yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3461no—
403C4—3461no—
404C4—6921no—
405C4—6921no—
406C4—6921no—
407C4C22×D69211yescharacteristic
408C4#1914972511yescharacteristic
409C2—42no—
410C2—42no—
411C2—13841no—
412C2—6921no—
413C2—6921no—
414C2D4×D69211yescharacteristic
415C2#2601228711yescharacteristic
416C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#442×C173C2
socleC22×C173C22×D4
hypercenter#442D173
hypocenterC173#3267
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×#19149698C2semidirect
3. #26012145C2semidirect
4. #26012136C2semidirect
5. #26011885C2semidirect
6. #26011750C2semidirect
7. #26011721C2semidirect
8. #26011706C2semidirect
9. #26011696C2semidirect
10. #26011654C2semidirect
11. #26011654C2semidirect
12. #26011643C2semidirect
13. #26011637C2semidirect
14. #26011564C2semidirect
15. #442×C173C2semidirect
18. C2×#14583825C22semidirect
22. C2×Dic692C22semidirect
23. C2×Dic692C22semidirect
28. C2×#14583814C22semidirect
31. C2×C8×D173C22semidirect
32. #19149679C22semidirect
39. #19149659C22semidirect
47. C4⋊#13063436C22semidirect
48. #19149602C22semidirect
49. #19149602C22semidirect
50. #19149602C22semidirect
53. #19149591C22semidirect
56. C4×D4×C173C22semidirect
57. #106×C173C22semidirect
59. #102×C173C22semidirect
60. #97×C173C22semidirect
115. D8×C173D4semidirect
208. C173#3267semidirect
223. #442D173semidirect
256. C4×D4C2×D173semidirect
259. #106C2×D173semidirect
266. #102C2×D173semidirect
269. #97C2×D173semidirect
306. D8D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×#191496982
#260121452
#260121362
#260118852
#260117502
#260117212
#260117062
#260116962
#260116542
#260116432
#260116372
#260115642
#442×C1732
#3267173