Finite Group Databasebeta

← #36363103

Subgroups — #36363103

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×#19149703C211yesmaximal, characteristic, complemented
2C2×#19149700C211yesmaximal, characteristic, complemented
3#26012059C211yesmaximal, characteristic, complemented
4#26011970C211yesmaximal, characteristic, complemented
5#26011716C211yesmaximal, characteristic, complemented
6#26011710C211yesmaximal, characteristic, complemented
7#26011708C211yesmaximal, characteristic, complemented
8#26011692C211yesmaximal, characteristic, complemented
9#26011690C211yesmaximal, characteristic, complemented
10#26011686C211yesmaximal, characteristic, complemented
11#26011684C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149703—24no—
20#19149700—24no—
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583815C2211yescharacteristic
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic
25C2×#14583811C2211yescharacteristic
26C2×#14583811C2211yescharacteristic, complemented
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149676C2211yescharacteristic
30#19149675C2211yescharacteristic
31#19149668C2211yescharacteristic
32#19149652—21no—
33#19149652—21no—
34#19149649—21no—
35#19149649—21no—
36C4×Dic346C2211yescharacteristic, complemented
37#19149644C2211yescharacteristic, complemented
38D4×Dic173C2211yescharacteristic, complemented
39C4×C22⋊D173—21no—
40#19149639—21no—
41#19149636C2211yescharacteristic
42#19149635—21no—
43C2×#14583805—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×SD16×C173C2211yescharacteristic, complemented
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic
61C23×D173—21no—
62#14583827—22no—
63#14583827—22no—
64#14583825—22no—
65#14583825D414yescomplemented
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—21no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76D4×D173—22no—
77D4×D173—22no—
78C22×Dic173—21no—
79C22×Dic173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83#14583816—22no—
84#14583815—22no—
85#14583815—22no—
86#14583814—22no—
87#14583811—22no—
88#14583811—22no—
89#14583809—22no—
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C8×D173—22no—
93C4⋊Dic173C2311yescharacteristic
94#14583805—21no—
95#14583805—21no—
96#14583805—21no—
97#14583805D411yescharacteristic
98#14583805D411yescharacteristic
99C22⋊Dic173—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803D411yescharacteristic
103#14583803D411yescharacteristic
104C4×Dic173—21no—
105C4×Dic173—21no—
106C4×Dic173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110C22×C4×C173—21no—
111SD16×C173—22no—
112D8×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173—21no—
121C22×D173—21no—
122C22×D173—42no—
123C22×D173—81no—
124C22×D173—81no—
125C22×D173C2×D411yescharacteristic
126Dic346—21no—
127Dic346C2×D412yes—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—42no—
133D692—81no—
134D692—81no—
135C22⋊D173—24no—
136C22⋊D173—42no—
137C22⋊D173—81no—
138C22⋊D173—81no—
139C2×Dic173—21no—
140C2×Dic173—24no—
141C2×Dic173—21no—
142C2×Dic173—81no—
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—42no—
147C4×D173—81no—
148C4×D173C2×D412yes—
149C4×D173#4112yes—
150#13063436—21no—
151#13063436—21no—
152C23×C173—21no—
153C23×C173—21no—
154Q8×C173—21no—
155Q8×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173C2×D412yes—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—81no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173—21no—
168C22×C173—22no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
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—21no—
186C4×C173—21no—
187C4×C173—41no—
188C4×C173—41no—
189C4×C173C22×D411yescharacteristic
190C4×C173C2×#4111yescharacteristic
191Dic173—22no—
192Dic173—21no—
193Dic173—21no—
194Dic173—41no—
195Dic173#12112yes—
196C2×C173—42no—
197C2×C173—81no—
198C2×C173C2×#12111yescharacteristic
199C2×C173#57611yescharacteristic
200C2×C173#57711yescharacteristic
201D173—22no—
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#464—1731no—
212#463—1731no—
213#463D17311yescharacteristic, complemented
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×Q16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16C2×D17311yescharacteristic, complemented
231C2×D8—1731no—
232C2×D8—1731no—
233C2×D8C2×D17311yescharacteristic
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×Q8—1731no—
246C4×D4—3461no—
247C4×D4—3461no—
248C4×D4—1731no—
249C4×D4C2×D17311yescharacteristic, complemented
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×Q8—1731no—
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×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—21no—
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292Q16—3462no—
293SD16—22no—
294SD16—3462no—
295SD16—3462no—
296SD16—3462no—
297D8—22no—
298D8—3462no—
299D8—3462no—
300M16—3462no—
301C2×C8—3462no—
302C2×C8—1731no—
303C2×C8—1731no—
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C4⋊C4—3461no—
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—21no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—1731no—
320C22⋊C4—1731no—
321C42—3461no—
322C42—3461no—
323C42—1731no—
324C42C22×D17311yescharacteristic
325C23—21no—
326C23—21no—
327C23—3461no—
328C23—3461no—
329C23—13841no—
330C23—13841no—
331C23—1731no—
332C23—6922no—
333Q8—21no—
334Q8—21no—
335Q8—3461no—
336Q8—1732no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—3464no—
341D4—3461no—
342D4—3461no—
343D4—3461no—
344D4—3461no—
345D4—13841no—
346D4—13841no—
347D4—13841no—
348D4—13841no—
349D4—6922no—
350D4—6922no—
351D4D4×D17312yes—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—81no—
355C2×C4—3461no—
356C2×C4—3464no—
357C2×C4—3461no—
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—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—3461no—
380C22—3461no—
381C22—13844no—
382C22—1732no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22—6921no—
388C22—6921no—
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—3462no—
395C4—3461no—
396C4—3461no—
397C4—1732no—
398C4—6921no—
399C4C2×D4×D17311yescharacteristic
400C4#41×D17311yescharacteristic
401C2—42no—
402C2—81no—
403C2—3462no—
404C2—13841no—
405C2—13841no—
406C2#121×D17311yescharacteristic
407C2#2601229511yescharacteristic
408C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
hypercenter#463D173
hypocenterC173#3045
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×#19149700C2semidirect
3. #26012059C2semidirect
4. #26011970C2semidirect
5. #26011716C2semidirect
6. #26011710C2semidirect
7. #26011708C2semidirect
8. #26011692C2semidirect
9. #26011690C2semidirect
10. #26011686C2semidirect
11. #26011684C2semidirect
12. #26011516C2semidirect
13. #26011491C2semidirect
14. #102×D173C2semidirect
15. #463×C173C2semidirect
21. C2×#14583816C22semidirect
23. C2×#14583815C22semidirect
26. C2×#14583811C22semidirect
36. C4×Dic346C22semidirect
37. #19149644C22semidirect
38. D4×Dic173C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149603C22semidirect
50. #19149601C22semidirect
53. #19149599C22semidirect
54. C2×SD16×C173C22semidirect
57. C4×D4×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
65. #14583825D4semidirect
204. C173#3045semidirect
213. #463D173semidirect
230. C2×SD16C2×D173semidirect
249. C4×D4C2×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×#191497032
C2×#191497002
#260120592
#260119702
#260117162
#260117102
#260117082
#260116922
#260116902
#260116862
#260116842
#260115162
#260114912
#102×D1732
#463×C1732
#3045173