Finite Group Databasebeta

← #36363105

Subgroups — #36363105

Order 22144.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149706C211yesmaximal, characteristic, complemented
2C2×#19149702C211yesmaximal, characteristic, complemented
3#26012059C211yesmaximal, characteristic, complemented
4#26011973C211yesmaximal, characteristic, complemented
5#26011715C211yesmaximal, characteristic, complemented
6#26011709C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011691C211yesmaximal, characteristic, complemented
9#26011689C211yesmaximal, characteristic, complemented
10#26011687C211yesmaximal, characteristic, complemented
11#26011685C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#464×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583827C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149706—24no—
20#19149702—24no—
21C2×#14583813C2211yescharacteristic
22C2×#14583813C2211yescharacteristic, complemented
23C2×#14583813C2211yescharacteristic, complemented
24C2×D1384C2211yescharacteristic
25C2×D1384C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149676C2211yescharacteristic
30#19149675C2211yescharacteristic
31#19149668C2211yescharacteristic
32#19149652—21no—
33#19149652—21no—
34#19149649—21no—
35#19149649—21no—
36Q8×Dic173C2211yescharacteristic, complemented
37C4×D692—21no—
38C4×D692C2211yescharacteristic, complemented
39#19149641—21no—
40#19149641C2211yescharacteristic, complemented
41#19149636—21no—
42#19149636C2211yescharacteristic
43C4⋊C4×D173—21no—
44C4⋊C4×D173C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149608C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic
52#19149599C2211yescharacteristic
53#19149599C2211yescharacteristic, complemented
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—
62#14583827—22no—
63#14583827—22no—
64#14583827—22no—
65#14583827D414yescomplemented
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—21no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76D4×D173—22no—
77D4×D173—22no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173C2311yescharacteristic
84#14583813—22no—
85#14583813—22no—
86#14583813—22no—
87D1384—22no—
88D1384—22no—
89#14583811—22no—
90#14583809—22no—
91C2×#13063436C2311yescharacteristic
92C2×#13063436C2311yescharacteristic
93C8×D173—22no—
94C4⋊Dic173—21no—
95C4⋊Dic173C2311yescharacteristic
96#14583805—21no—
97#14583805D411yescharacteristic
98#14583805D411yescharacteristic
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803D411yescharacteristic
104#14583803D411yescharacteristic
105C4×Dic173—21no—
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—21no—
122C22×D173—21no—
123C22×D173—42no—
124C22×D173—81no—
125C22×D173—81no—
126C22×D173C2×D411yescharacteristic
127D692—24no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—42no—
134D692—81no—
135D692—81no—
136D692C2×D412yes—
137C22⋊D173—42no—
138C22⋊D173—81no—
139C22⋊D173—81no—
140C2×Dic173—21no—
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C4×D173—24no—
145C4×D173—42no—
146C4×D173—81no—
147C4×D173—81no—
148C4×D173—81no—
149C4×D173C2×D412yes—
150C4×D173#4112yes—
151#13063436—21no—
152#13063436—21no—
153C23×C173—21no—
154Q8×C173—21no—
155Q8×C173—21no—
156Q8×C173—21no—
157Q8×C173C2×D412yes—
158D4×C173—21no—
159D4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173—21no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173C22×D411yescharacteristic
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—22no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C2×D173#12112yes—
185C4×C173—22no—
186C4×C173—21no—
187C4×C173—21no—
188C4×C173—41no—
189C4×C173—41no—
190C4×C173C22×D411yescharacteristic
191C4×C173C2×#4111yescharacteristic
192Dic173—21no—
193Dic173—21no—
194Dic173—41no—
195Dic173#12112yes—
196C2×C173—81no—
197C2×C173C2×#12111yescharacteristic
198C2×C173#57611yescharacteristic
199C2×C173#57711yescharacteristic
200D173—22no—
201D173—42no—
202D173—81no—
203D173—81no—
204C173#304511yescharacteristic, complemented, Sylow
205#3045—1731nomaximal, Sylow
206C2×AGL(1,ℤ/8)—1731no—
207C2×#135—1731no—
208#535—1731no—
209#520—1731no—
210#464—1731no—
211#464D17311yescharacteristic, complemented
212#463—1731no—
213#463—1731no—
214#460—1731no—
215#460—1731no—
216#459—1731no—
217#459—1731no—
218#424—1731no—
219#417—1731no—
220C2×#102—1731no—
221C2×#41—1731no—
222C2×#41—1731no—
223C22×D4—1731no—
224AGL(1,ℤ/8)—3464no—
225#135—3464no—
226C2×Q16C2×D17311yescharacteristic, complemented
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16C2×D17311yescharacteristic
231C2×D8—1731no—
232C2×D8—1731no—
233C2×D8—1731no—
234C2×M16—1731no—
235C22×C8—1731no—
236#127—1731no—
237#127—1731no—
238#124—1731no—
239#124C2×D17311yescharacteristic
240#121—3461no—
241#121—3461no—
242#120—3461no—
243#120—3461no—
244C4×Q8—1731no—
245C4×Q8C2×D17311yescharacteristic, complemented
246C4×D4—3461no—
247C4×D4—3461no—
248C4×D4—1731no—
249C4×D4—1731no—
250#117—3461no—
251#117—1731no—
252C2×C4⋊C4—3461no—
253C2×C4⋊C4—1731no—
254C4⋊C8—1731no—
255C4⋊C8—1731no—
256C4⋊C8—1731no—
257C4⋊C8C2×D17311yescharacteristic, complemented
258#103—1731no—
259#103C2×D17311yescharacteristic, complemented
260#102—1731no—
261#102—1731no—
262#102—1731no—
263#102—1731no—
264#102—1731no—
265#102C2×D17311yescharacteristic
266C24—3461no—
267#41—3462no—
268#41—3462no—
269#41—3462no—
270#41—1734no—
271C2×Q8C22×D17311yescharacteristic
272C2×Q8C22×D17311yescharacteristic
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3462no—
276C2×D4—3462no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—1731no—
283C2×D4—1731no—
284C2×D4—1731no—
285C2×D4C22×D17311yescharacteristic
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292Q16—22no—
293SD16—22no—
294SD16—3462no—
295SD16—3462no—
296SD16—3462no—
297D8—3462no—
298D8—3462no—
299D8—3462no—
300M16—3462no—
301C2×C8—3462no—
302C2×C8—1731no—
303C2×C8—1731no—
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4—1731no—
312C4⋊C4C22×D17311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—1731no—
320C22⋊C4—1731no—
321C42—21no—
322C42—3461no—
323C42—1731no—
324C42C22×D17311yescharacteristic
325C23—21no—
326C23—3461no—
327C23—3461no—
328C23—3461no—
329C23—13841no—
330C23—13841no—
331C23—1731no—
332C23—6922no—
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8D4×D17312yes—
337D4—21no—
338D4—21no—
339D4—3464no—
340D4—3461no—
341D4—3461no—
342D4—3461no—
343D4—3461no—
344D4—3461no—
345D4—13841no—
346D4—13841no—
347D4—13841no—
348D4—13841no—
349D4—1732no—
350D4—6922no—
351D4—6922no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—3464no—
356C2×C4—3461no—
357C2×C4—13841no—
358C2×C4—13841no—
359C2×C4—13841no—
360C2×C4—1731no—
361C2×C4—1731no—
362C2×C4—1731no—
363C2×C4—1732no—
364C2×C4—1732no—
365C2×C4—6922no—
366C2×C4D4×D17311yescharacteristic
367C2×C4D4×D17311yescharacteristic
368C2×C4C23×D17311yescharacteristic
369C8—21no—
370C8—21no—
371C8—3461no—
372C8—3461no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—3461no—
377C22—3461no—
378C22—3462no—
379C22—13844no—
380C22—1732no—
381C22—6921no—
382C22—6921no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22—6921no—
388C22—6921no—
389C22C2×D4×D17311yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—3461no—
396C4—3461no—
397C4—1732no—
398C4—6921no—
399C4C2×D4×D17311yescharacteristic
400C4#41×D17311yescharacteristic
401C2—81no—
402C2—3462no—
403C2—13841no—
404C2—13841no—
405C2—6922no—
406C2#121×D17311yescharacteristic
407C2#2601229311yescharacteristic
408C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
hypercenter#464D173
hypocenterC173#3045
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×#19149706C2semidirect
2. C2×#19149702C2semidirect
3. #26012059C2semidirect
4. #26011973C2semidirect
5. #26011715C2semidirect
6. #26011709C2semidirect
7. #26011703C2semidirect
8. #26011691C2semidirect
9. #26011689C2semidirect
10. #26011687C2semidirect
11. #26011685C2semidirect
12. #26011516C2semidirect
13. #26011491C2semidirect
14. #102×D173C2semidirect
15. #464×C173C2semidirect
22. C2×#14583813C22semidirect
23. C2×#14583813C22semidirect
25. C2×D1384C22semidirect
36. Q8×Dic173C22semidirect
38. C4×D692C22semidirect
40. #19149641C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149603C22semidirect
50. #19149601C22semidirect
53. #19149599C22semidirect
54. C2×Q16×C173C22semidirect
57. C4×Q8×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
65. #14583827D4semidirect
204. C173#3045semidirect
211. #464D173semidirect
226. C2×Q16C2×D173semidirect
245. C4×Q8C2×D173semidirect
257. C4⋊C8C2×D173semidirect
259. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×#191497022
#260120592
#260119732
#260117152
#260117092
#260117032
#260116912
#260116892
#260116872
#260116852
#260115162
#260114912
#102×D1732
#464×C1732
#3045173