Finite Group Databasebeta

← #464×D173 #36363003

Subgroups — #464×D173 #36363003

Order 22144.

500 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×D173C211yesmaximal, characteristic, complemented
2C2×SD16×D173C211yesmaximal, characteristic
3#124×D173C211yesmaximal, characteristic
4C4×Q8×D173C211yesmaximal, characteristic, complemented
5#26011716C211yesmaximal, characteristic, complemented
6#26011715C211yesmaximal, characteristic, complemented
7#26011714C211yesmaximal, characteristic, complemented
8#26011713C211yesmaximal, characteristic, complemented
9#26011712C211yesmaximal, characteristic, complemented
10#26011711C211yesmaximal, characteristic, complemented
11#26011710C211yesmaximal, characteristic, complemented
12C4⋊C8×D173C211yesmaximal, characteristic, complemented
13#103×D173C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic
15#464×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19C2×Dic692C2211yescharacteristic, complemented
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583816C2211yescharacteristic, complemented
22Q16×D173—24no—
23C2×#14583815C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583813C2211yescharacteristic
26SD16×D173—24no—
27C2×C8×D173C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149668C2211yescharacteristic
30#19149667C2211yescharacteristic
31#19149666C2211yescharacteristic
32#19149665—21no—
33#19149665—21no—
34Q8×Dic173C2211yescharacteristic, complemented
35C4×Dic346—21no—
36C4×Dic346C2211yescharacteristic, complemented
37#19149644—21no—
38#19149644C2211yescharacteristic, complemented
39C4⋊C4×D173—21no—
40C4⋊C4×D173C2211yescharacteristic
41C22⋊C4×D173—21no—
42C22⋊C4×D173—21no—
43C42×D173—21no—
44C42×D173C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149608C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic
52#19149600C2211yescharacteristic
53#19149599C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57C4×Q8×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic
61C23×D173—21no—
62C2×Dic346—21no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—22no—
68Q8×D173—22no—
69Q8×D173D414yes—
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72D4×D173—22no—
73D4×D173—22no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173D411yescharacteristic
78C2×C4×D173D411yescharacteristic
79C2×C4×D173C2311yescharacteristic
80Dic692—22no—
81#14583816—22no—
82#14583816—22no—
83#14583815—22no—
84#14583814—22no—
85#14583813—22no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C8×D173—22no—
89C8×D173—22no—
90C4⋊Dic173—21no—
91C4⋊Dic173C2311yescharacteristic
92#14583805—21no—
93#14583805—21no—
94#14583805—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101C4×Dic173—21no—
102C4×Dic173—21no—
103C4×Dic173—21no—
104C4×Dic173D411yescharacteristic
105C4×Dic173D411yescharacteristic
106C4×Dic173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110Q16×C173—22no—
111SD16×C173—22no—
112C2×C8×C173C2311yescharacteristic
113C2×C8×C173C2311yescharacteristic
114C4⋊C4×C173—21no—
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C42×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173—21no—
121C22×D173—42no—
122C22×D173—42no—
123C22×D173—42no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—24no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130Dic346C2×D412yes—
131D692—21no—
132D692—21no—
133C22⋊D173—42no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—24no—
141C4×D173—22no—
142C4×D173—22no—
143C4×D173—42no—
144C4×D173—42no—
145C4×D173C2×D412yes—
146C4×D173#4112yes—
147#13063436—21no—
148#13063436—21no—
149C23×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173C2×D412yes—
154D4×C173—21no—
155D4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2411yescharacteristic
162C8×C173—21no—
163C8×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—84no—
172C2×D173#12112yes—
173C2×D173#13512yes—
174C2×D173#13712yes—
175C4×C173—22no—
176C4×C173—21no—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173C22×D411yescharacteristic
181C4×C173C2×#4111yescharacteristic
182Dic173—22no—
183Dic173—21no—
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187Dic173#12112yes—
188C2×C173—81no—
189C2×C173C2×#12111yescharacteristic
190C2×C173C2×#13511yescharacteristic
191C2×C173C2×#13711yescharacteristic
192D173—81no—
193D173#46414yesdirect product factor
194C173C2×#46411yescharacteristic, complemented, Sylow
195C2×#464—1731nomaximal, Sylow
196C22×Q16—1731no—
197C22×SD16—1731no—
198C2×#124—1731no—
199C2×C4×Q8—1731no—
200#464—1731no—
201#464—1731no—
202#464—1731no—
203#464—1731no—
204#464—1731no—
205#464—1731no—
206#464—1731no—
207#464D17311yescharacteristic, direct product factor
208C2×C4⋊C8—1731no—
209C2×#103—1731no—
210C2×#102—1731no—
211C22×Q8—1731no—
212C22×Q8—1731no—
213C22×D4—1731no—
214C2×Q16—3464no—
215C2×Q16—1731no—
216C2×Q16—1731no—
217C2×Q16—1731no—
218C2×Q16C2×D17311yescharacteristic, complemented
219C2×SD16—3464no—
220C2×SD16—1731no—
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×SD16C2×D17311yescharacteristic
224C22×C8—1731no—
225C22×C8—1731no—
226#124—3461no—
227#124—3461no—
228#124—1731no—
229#124—1731no—
230#124—1731no—
231#124C2×D17311yescharacteristic
232C4×Q8—3461no—
233C4×Q8—3461no—
234C4×Q8—1731no—
235C4×Q8—1731no—
236C4×Q8—1731no—
237C4×Q8C2×D17311yescharacteristic, complemented
238C2×C4⋊C4—3461no—
239C2×C4⋊C4—1731no—
240C2×C22⋊C4—3461no—
241C2×C22⋊C4—3461no—
242C2×C42—3461no—
243C2×C42—1731no—
244C4⋊C8—1731no—
245C4⋊C8—1731no—
246C4⋊C8—1731no—
247C4⋊C8C2×D17311yescharacteristic, complemented
248#103—1731no—
249#103—1731no—
250#103—1731no—
251#103C2×D17311yescharacteristic, complemented
252#102—1731no—
253#102—1731no—
254#102—1731no—
255#102C2×D17311yescharacteristic
256C24—3461no—
257C2×Q8—3461no—
258C2×Q8—3461no—
259C2×Q8—3462no—
260C2×Q8—3462no—
261C2×Q8—3462no—
262C2×Q8—1734no—
263C2×Q8—1731no—
264C2×Q8—1731no—
265C2×Q8C22×D17311yescharacteristic
266C2×Q8C22×D17311yescharacteristic
267C2×D4—3462no—
268C2×D4—3462no—
269C2×D4—3461no—
270C2×D4—1731no—
271C2×D4C22×D17311yescharacteristic
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—1731no—
276C22×C4—1731no—
277C22×C4—1731no—
278Q16—22no—
279Q16—3462no—
280Q16—3462no—
281Q16—3462no—
282SD16—22no—
283SD16—3462no—
284SD16—3462no—
285SD16—3462no—
286C2×C8—3462no—
287C2×C8—3462no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8C22×D17311yescharacteristic
291C2×C8C22×D17311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—3461no—
297C4⋊C4—1731no—
298C4⋊C4C22×D17311yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C42—21no—
308C42—3461no—
309C42—3461no—
310C42—3461no—
311C42—1731no—
312C42—1731no—
313C42—1731no—
314C42C22×D17311yescharacteristic
315C23—21no—
316C23—3461no—
317C23—1731no—
318C23—6922no—
319C23—6922no—
320C23—6922no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—3464no—
325Q8—3461no—
326Q8—3461no—
327Q8—3461no—
328Q8—1732no—
329Q8—6922no—
330Q8D4×D17312yes—
331D4—21no—
332D4—21no—
333D4—3461no—
334D4—3461no—
335D4—6922no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—3464no—
340C2×C4—3462no—
341C2×C4—3462no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—1731no—
346C2×C4—1731no—
347C2×C4—1731no—
348C2×C4—1732no—
349C2×C4—1732no—
350C2×C4—6922no—
351C2×C4—6922no—
352C2×C4D4×D17311yescharacteristic
353C2×C4D4×D17311yescharacteristic
354C2×C4C23×D17311yescharacteristic
355C8—21no—
356C8—21no—
357C8—3461no—
358C8—3461no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—13844no—
363C22—1732no—
364C22—1732no—
365C22—1732no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22C2×D4×D17311yescharacteristic
370C4—22no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3462no—
376C4—3461no—
377C4—3461no—
378C4—1732no—
379C4—6921no—
380C4—6921no—
381C4C2×D4×D17311yescharacteristic
382C4#41×D17311yescharacteristic
383C2—81no—
384C2—13841no—
385C2—1734no—
386C2#121×D17311yescharacteristic
387C2#135×D17311yescharacteristic
388C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#464D173
hypocenterC173C2×#464
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D173C2semidirect
4. C4×Q8×D173C2semidirect
5. #26011716C2semidirect
6. #26011715C2semidirect
7. #26011714C2semidirect
8. #26011713C2semidirect
9. #26011712C2semidirect
10. #26011711C2semidirect
11. #26011710C2semidirect
12. C4⋊C8×D173C2semidirect
13. #103×D173C2semidirect
15. #464×C173C2semidirect
19. C2×Dic692C22semidirect
20. C2×#14583816C22semidirect
21. C2×#14583816C22semidirect
34. Q8×Dic173C22semidirect
36. C4×Dic346C22semidirect
38. #19149644C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149603C22semidirect
50. #19149602C22semidirect
54. C2×Q16×C173C22semidirect
57. C4×Q8×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
193. D173#464direct (N × H)
194. C173C2×#464semidirect
207. #464D173direct (N × H)
218. C2×Q16C2×D173semidirect
237. C4×Q8C2×D173semidirect
247. C4⋊C8C2×D173semidirect
251. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1732
C2×SD16×D1732
#124×D1732
C4×Q8×D1732
#260117162
#260117152
#260117142
#260117132
#260117122
#260117112
#260117102
C4⋊C8×D1732
#103×D1732
#102×D1732
#464×C1732
C2×#464173