Finite Group Databasebeta

← #36365082

Subgroups — #36365082

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×#19149698C211yesmaximal, characteristic, complemented
2#26012140C211yesmaximal, characteristic, complemented
3#26012135C211yesmaximal, characteristic, complemented
4#26011882C211yesmaximal, characteristic, complemented
5#26011749C211yesmaximal, characteristic, complemented
6#26011717C211yesmaximal, characteristic, complemented
7#26011704C211yesmaximal, characteristic, complemented
8#26011694C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011603C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic, complemented
18#19149698—24no—
19C2×Dic692C2211yescharacteristic, complemented
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23C2×#14583814C2211yescharacteristic, complemented
24C2×C8×D173C2211yescharacteristic, complemented
25#19149677C2211yescharacteristic, complemented
26#19149666C2211yescharacteristic
27#19149662—21no—
28#19149660—21no—
29#19149660—21no—
30#19149658C2211yescharacteristic
31#19149658C2211yescharacteristic
32#19149655C2211yescharacteristic, complemented
33#19149651—21no—
34#19149651—21no—
35#19149650C2211yescharacteristic
36#19149648—21no—
37D4×Dic173C2211yescharacteristic
38D4×Dic173C2211yescharacteristic, complemented
39C4×C22⋊D173C2211yescharacteristic
40C2×#14583805—21no—
41C2×C22⋊Dic173—21no—
42#19149611C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149604C2211yescharacteristic, complemented
46#19149603C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic, complemented
50C22⋊#13063436C2211yescharacteristic
51#19149591C2211yescharacteristic, complemented
52#19149589C2211yescharacteristic, complemented
53C2×#41×C173C2211yescharacteristic
54C22×D4×C173C2211yescharacteristic
55AGL(1,ℤ/8)×C173—24no—
56C2×SD16×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic
58C2×D8×C173C2211yescharacteristic
59C2×D8×C173C2211yescharacteristic
60C2×M16×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—22no—
63#14583825—41no—
64#14583825—41no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C22×Dic173—21no—
73C22×Dic173—21no—
74C22×Dic173—21no—
75C2×C4×D173C2311yescharacteristic
76Dic692—22no—
77#14583816—41no—
78#14583814—22no—
79#14583814—22no—
80#14583814—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—22no—
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91#14583805D411yescharacteristic
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99C22⋊Dic173D411yescharacteristic
100#14583803—21no—
101#14583803D411yescharacteristic
102C4×Dic173C2311yescharacteristic
103C24×C173—21no—
104#41×C173—22no—
105#41×C173—41no—
106C2×Q8×C173C2311yescharacteristic
107C2×D4×C173—22no—
108C2×D4×C173—21no—
109C2×D4×C173—21no—
110C2×D4×C173—41no—
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116SD16×C173—22no—
117SD16×C173—22no—
118D8×C173—22no—
119D8×C173D414yescomplemented
120M16×C173—22no—
121C2×C8×C173C2311yescharacteristic
122C2×C8×C173C2311yescharacteristic
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128C22⋊D173—41no—
129C22⋊D173—42no—
130C22⋊D173—42no—
131C22⋊D173—81no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C2×Dic173—42no—
138C2×Dic173—42no—
139C2×Dic173—81no—
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C4×D173—21no—
144C4×D173—21no—
145#13063436—21no—
146#13063436—21no—
147C23×C173—21no—
148C23×C173—21no—
149C23×C173—21no—
150C23×C173—42no—
151C23×C173—42no—
152C23×C173—81no—
153C23×C173C2×D411yescharacteristic
154Q8×C173—21no—
155Q8×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—42no—
161D4×C173—42no—
162D4×C173C2×D412yes—
163D4×C173C2×D412yes—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—42no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—21no—
173C22×C173—22no—
174C22×C173—22no—
175C22×C173—21no—
176C22×C173—21no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—84no—
185C22×C173C22×D411yescharacteristic
186C2×D173—21no—
187C2×D173—21no—
188C2×D173—21no—
189C4×C173—21no—
190C4×C173—41no—
191C4×C173C22×D411yescharacteristic
192C4×C173H(5,𝔽2)11yescharacteristic
193Dic173—21no—
194Dic173—21no—
195Dic173—21no—
196Dic173—41no—
197Dic173—41no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—41no—
201C2×C173—42no—
202C2×C173—42no—
203C2×C173—81no—
204C2×C173D4211yescharacteristic
205C2×C173#57511yescharacteristic
206C2×C173#57711yescharacteristic
207D173—41no—
208C173#326711yescharacteristic, complemented, Sylow
209#3267—1731nomaximal, Sylow
210C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
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#442—1731no—
224#435—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C2×#41C2×D17311yescharacteristic
228C22×D4C2×D17311yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—3464no—
231C2×Q16—1731no—
232C2×Q16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×SD16C2×D17311yescharacteristic
237C2×SD16C2×D17311yescharacteristic
238C2×D8C2×D17311yescharacteristic
239C2×D8C2×D17311yescharacteristic
240C2×M16C2×D17311yescharacteristic
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×D4—1731no—
256C4×D4—1731no—
257C2×C4⋊C4—3461no—
258C2×C22⋊C4—3461no—
259#106—1731no—
260C4⋊C8—1731no—
261#103—1731no—
262#103—1731no—
263#103—1731no—
264#102—1731no—
265#102—1731no—
266#102—1731no—
267C22⋊C8—1731no—
268C22⋊C8—1731no—
269#97—1731no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—3462no—
274#41—3462no—
275#41—6921no—
276#41—6921no—
277C2×Q8—1731no—
278C2×Q8—1731no—
279C2×Q8C22×D17311yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—1731no—
288C2×D4C22×D17311yescharacteristic
289C2×D4C22×D17311yescharacteristic
290C2×D4C22×D17311yescharacteristic
291C22×C4—21no—
292C22×C4—3461no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4C22×D17311yescharacteristic
298Q16—3462no—
299Q16—6921no—
300SD16—22no—
301SD16—22no—
302SD16—3462no—
303SD16—3462no—
304SD16—6921no—
305D8—22no—
306D8C22⋊D17314yescomplemented
307M16—22no—
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⋊C4—1731no—
320C4⋊C4—1731no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—1731no—
330C22⋊C4—1731no—
331C42—1731no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1731no—
339C23D4×D17311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3461no—
343Q8—3461no—
344Q8—3461no—
345Q8—3461no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—13841no—
353D4—6921no—
354D4—6922no—
355D4—6922no—
356D4C2×C22⋊D17312yes—
357D4C2×C22⋊D17312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—3461no—
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—1731no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4—6922no—
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—3461no—
379C8—3461no—
380C8C2×C22⋊D17312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3461no—
395C22—3461no—
396C22—3461no—
397C22C2×D4×D17311yescharacteristic
398C4—21no—
399C4—41no—
400C4—3461no—
401C4—3461no—
402C4—3461no—
403C4—6921no—
404C4—6921no—
405C4—6921no—
406C4—6921no—
407C4C22×C22⋊D17311yescharacteristic
408C4#1914972411yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—6921no—
414C2D4×C22⋊D17311yescharacteristic
415C2#2601228711yescharacteristic
416C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3267
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19149698C2semidirect
2. #26012140C2semidirect
3. #26012135C2semidirect
4. #26011882C2semidirect
5. #26011749C2semidirect
6. #26011717C2semidirect
7. #26011704C2semidirect
8. #26011694C2semidirect
9. #26011653C2semidirect
10. #26011650C2semidirect
11. #26011642C2semidirect
12. #26011633C2semidirect
13. #26011603C2semidirect
14. #26011560C2semidirect
15. C2×AGL(1,ℤ/8)×C173C2semidirect
17. C2×#14583825C22semidirect
19. C2×Dic692C22semidirect
20. C2×#14583816C22semidirect
23. C2×#14583814C22semidirect
24. C2×C8×D173C22semidirect
25. #19149677C22semidirect
32. #19149655C22semidirect
38. D4×Dic173C22semidirect
42. #19149611C22semidirect
43. #19149608C22semidirect
44. #19149604C22semidirect
45. #19149604C22semidirect
46. #19149603C22semidirect
49. #19149600C22semidirect
51. #19149591C22semidirect
52. #19149589C22semidirect
119. D8×C173D4semidirect
208. C173#3267semidirect
210. C2×AGL(1,ℤ/8)D173semidirect
306. D8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496982
#260121402
#260121352
#260118822
#260117492
#260117172
#260117042
#260116942
#260116532
#260116502
#260116422
#260116332
#260116032
#260115602
C2×AGL(1,ℤ/8)×C1732
#3267173