Finite Group Databasebeta

← #36365168

Subgroups — #36365168

Order 22144.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic, complemented
2#26012135C211yesmaximal, characteristic, complemented
3#26012129C211yesmaximal, characteristic, complemented
4#26011872C211yesmaximal, characteristic, complemented
5#26011737C211yesmaximal, characteristic, complemented
6#26011734C211yesmaximal, characteristic, complemented
7#26011708C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13D8×Dic173C211yesmaximal, characteristic, complemented
14#26011567C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic, complemented
18#19149703—24no—
19C2×#14583816C2211yescharacteristic, complemented
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic, complemented
22C2×#14583811C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24C2×#14583809C2211yescharacteristic, complemented
25#19149668C2211yescharacteristic, complemented
26#19149666C2211yescharacteristic, complemented
27#19149662—21no—
28#19149660—21no—
29#19149658C2211yescharacteristic, complemented
30#19149655C2211yescharacteristic, complemented
31#19149653C2211yescharacteristic
32#19149651—21no—
33#19149651—21no—
34#19149650C2211yescharacteristic
35#19149648—21no—
36#19149647—21no—
37D4×Dic173C2211yescharacteristic, complemented
38D4×Dic173C2211yescharacteristic, complemented
39C4×C22⋊D173C2211yescharacteristic
40C2×C22⋊Dic173—21no—
41C2×#14583803—21no—
42#19149613C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50C22⋊#13063436C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C8×Dic173C2211yescharacteristic, complemented
53C2×#41×C173C2211yescharacteristic
54C22×D4×C173C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C173—24no—
56C2×SD16×C173C2211yescharacteristic, complemented
57C2×SD16×C173C2211yescharacteristic, complemented
58C2×D8×C173C2211yescharacteristic
59C2×D8×C173C2211yescharacteristic
60C2×M16×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583825—22no—
63#14583825—41no—
64C2×Dic346C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71D4×D173—22no—
72D4×D173—41no—
73C22×Dic173—21no—
74C22×Dic173—21no—
75C22×Dic173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583816—41no—
78#14583815—22no—
79#14583814—22no—
80#14583811—22no—
81#14583811—41no—
82#14583809—22no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173D411yescharacteristic
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
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×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173—81no—
127C22×D173C2×D411yescharacteristic
128Dic346—21no—
129Dic346—21no—
130D692—21no—
131D692—21no—
132C22⋊D173—41no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—81no—
136C22⋊D173—81no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—42no—
141C2×Dic173—81no—
142C2×Dic173—81no—
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148#13063436—21no—
149#13063436—21no—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—21no—
153C23×C173—42no—
154C23×C173—42no—
155C23×C173—81no—
156C23×C173C2×D411yescharacteristic
157Q8×C173—21no—
158Q8×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—42no—
164D4×C173—42no—
165D4×C173C2×D412yes—
166D4×C173C2×D412yes—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—42no—
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173C2×D412yes—
175C22×C173—21no—
176C22×C173—22no—
177C22×C173—22no—
178C22×C173—21no—
179C22×C173—21no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—84no—
188C22×C173C22×D411yescharacteristic
189C2×D173—21no—
190C2×D173—21no—
191C2×D173—21no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—41no—
195C2×D173—84no—
196C4×C173—21no—
197C4×C173—41no—
198C4×C173C22×D411yescharacteristic
199C4×C173H(5,𝔽2)11yescharacteristic
200Dic173—21no—
201Dic173—21no—
202Dic173—21no—
203Dic173—41no—
204Dic173—41no—
205Dic173—41no—
206C2×C173—42no—
207C2×C173—41no—
208C2×C173—42no—
209C2×C173—81no—
210C2×C173D4211yescharacteristic
211C2×C173#57511yescharacteristic
212C2×C173#57611yescharacteristic
213D173—41no—
214D173—81no—
215C173#327311yescharacteristic, complemented, Sylow
216#3273—1731nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1731no—
218C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
219#546—1731no—
220#546—1731no—
221#495—1731no—
222#468—1731no—
223#468—1731no—
224#463—1731no—
225#463—1731no—
226#452—1731no—
227#452—1731no—
228#447—1731no—
229#447—1731no—
230C4×D8—1731no—
231#436—1731no—
232C2×#41—1731no—
233C2×#41C2×D17311yescharacteristic
234C22×D4—1731no—
235C22×D4C2×D17311yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—3464no—
238C2×Q16—1731no—
239C2×SD16—1731no—
240C2×SD16—1731no—
241C2×SD16C2×D17311yescharacteristic, complemented
242C2×SD16C2×D17311yescharacteristic, complemented
243C2×D8—1731no—
244C2×D8—1731no—
245C2×D8C2×D17311yescharacteristic
246C2×D8C2×D17311yescharacteristic
247C2×M16—1731no—
248C2×M16C2×D17311yescharacteristic, complemented
249#124—1731no—
250#124—1731no—
251#123—3461no—
252#123—3461no—
253C22⋊Q8—1731no—
254C22⋊Q8—1731no—
255#121—3461no—
256#121—3461no—
257#121—1731no—
258#121—1731no—
259#120—3461no—
260#120—3461no—
261C4×D4—1731no—
262C4×D4—1731no—
263C4×D4—1731no—
264C2×C22⋊C4—3461no—
265C2×C22⋊C4—3461no—
266#107—1731no—
267C4⋊C8—1731no—
268#103—1731no—
269#103—1731no—
270#102—1731no—
271#102—1731no—
272#102—1731no—
273#102—1731no—
274C22⋊C8—1731no—
275C22⋊C8—1731no—
276C4×C8—1731no—
277C24—21no—
278C24—3461no—
279#41—22no—
280#41—41no—
281#41—3462no—
282#41—6921no—
283C2×Q8—1731no—
284C2×Q8C22×D17311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3462no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—3461no—
294C2×D4—1731no—
295C2×D4—1731no—
296C2×D4—6921no—
297C2×D4C22×D17311yescharacteristic
298C2×D4C22×D17311yescharacteristic
299C2×D4C22×D17311yescharacteristic
300C22×C4—21no—
301C22×C4—3461no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4—1731no—
305C22×C4C22×D17311yescharacteristic
306Q16—6921no—
307SD16—22no—
308SD16—22no—
309SD16—3462no—
310SD16—3462no—
311D8—22no—
312D8—3462no—
313D8—6921no—
314D8C22⋊D17314yescomplemented
315M16—22no—
316M16—3462no—
317C2×C8—1731no—
318C2×C8—1731no—
319C2×C8C22×D17311yescharacteristic
320C2×C8C22×D17311yescharacteristic
321C4⋊C4—3461no—
322C4⋊C4—3461no—
323C4⋊C4—1731no—
324C4⋊C4—1731no—
325C4⋊C4—1731no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—3461no—
335C22⋊C4—3461no—
336C22⋊C4—1731no—
337C22⋊C4—1731no—
338C42—1731no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—3461no—
346C23—13841no—
347C23—1731no—
348C23—6922no—
349C23—6922no—
350C23D4×D17311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3461no—
354Q8—3461no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—3461no—
362D4—3461no—
363D4—13841no—
364D4—13841no—
365D4—6921no—
366D4—6922no—
367D4—6922no—
368D4C2×C22⋊D17312yes—
369D4C2×C22⋊D17312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—3461no—
378C2×C4—3461no—
379C2×C4—13841no—
380C2×C4—13841no—
381C2×C4—1731no—
382C2×C4—1731no—
383C2×C4—1731no—
384C2×C4—6922no—
385C2×C4D4×D17311yescharacteristic
386C2×C4C23×D17311yescharacteristic
387C8—21no—
388C8—3461no—
389C8—3461no—
390C8C2×C22⋊D17312yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3461no—
405C22—3461no—
406C22—3461no—
407C22—13844no—
408C22—6921no—
409C22—6921no—
410C22—6921no—
411C22C2×D4×D17311yescharacteristic
412C4—21no—
413C4—41no—
414C4—3461no—
415C4—3461no—
416C4—3461no—
417C4—6921no—
418C4—6921no—
419C4—6921no—
420C4C22×C22⋊D17311yescharacteristic
421C4#1914972411yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—13841no—
427C2—6921no—
428C2D4×C22⋊D17311yescharacteristic
429C2#2601228711yescharacteristic
430C2#2601229511yescharacteristic
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#3273
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×#19149703C2semidirect
2. #26012135C2semidirect
3. #26012129C2semidirect
4. #26011872C2semidirect
5. #26011737C2semidirect
6. #26011734C2semidirect
7. #26011708C2semidirect
8. #26011704C2semidirect
9. #26011653C2semidirect
10. #26011650C2semidirect
11. #26011625C2semidirect
12. #26011623C2semidirect
13. D8×Dic173C2semidirect
14. #26011567C2semidirect
15. C2×AGL(1,ℤ/8)×C173C2semidirect
17. C2×D4×D173C22semidirect
19. C2×#14583816C22semidirect
20. C2×#14583815C22semidirect
21. C2×#14583814C22semidirect
24. C2×#14583809C22semidirect
25. #19149668C22semidirect
26. #19149666C22semidirect
29. #19149658C22semidirect
30. #19149655C22semidirect
37. D4×Dic173C22semidirect
38. D4×Dic173C22semidirect
42. #19149613C22semidirect
43. #19149608C22semidirect
44. #19149604C22semidirect
45. #19149603C22semidirect
46. #19149601C22semidirect
47. #19149600C22semidirect
48. #19149600C22semidirect
49. #19149600C22semidirect
50. C22⋊#13063436C22semidirect
51. #19149591C22semidirect
52. C8×Dic173C22semidirect
54. C22×D4×C173C22semidirect
56. C2×SD16×C173C22semidirect
57. C2×SD16×C173C22semidirect
60. C2×M16×C173C22semidirect
119. D8×C173D4semidirect
215. C173#3273semidirect
218. C2×AGL(1,ℤ/8)D173semidirect
235. C22×D4C2×D173semidirect
241. C2×SD16C2×D173semidirect
242. C2×SD16C2×D173semidirect
248. C2×M16C2×D173semidirect
314. D8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
#260121352
#260121292
#260118722
#260117372
#260117342
#260117082
#260117042
#260116532
#260116502
#260116252
#260116232
D8×Dic1732
#260115672
C2×AGL(1,ℤ/8)×C1732
#3273173