Finite Group Databasebeta

← #36365072

Subgroups — #36365072

Order 22144.

494 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×#19149711C211yesmaximal, characteristic, complemented
2C2×SD16×D173C211yesmaximal, characteristic, complemented
3#26012156C211yesmaximal, characteristic, complemented
4#26012134C211yesmaximal, characteristic
5#26011734C211yesmaximal, characteristic, complemented
6#26011729C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011693C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011645C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011628C211yesmaximal, characteristic, complemented
13#26011593C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#497×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic
20#19149711—24no—
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583815C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583813C2211yescharacteristic
25C2×#14583813C2211yescharacteristic
26SD16×D173—24no—
27C2×#14583809C2211yescharacteristic, complemented
28C2×C8×D173C2211yescharacteristic
29#19149678—21no—
30#19149661—21no—
31#19149657—21no—
32#19149657—21no—
33#19149655C2211yescharacteristic
34#19149655C2211yescharacteristic
35#19149655C2211yescharacteristic, complemented
36#19149652—21no—
37#19149650C2211yescharacteristic
38#19149649—21no—
39C4×Dic346C2211yescharacteristic, complemented
40C4×D692C2211yescharacteristic
41C4×D692C2211yescharacteristic
42#19149641—21no—
43C22⋊C4×D173—21no—
44#19149613C2211yescharacteristic, complemented
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149604C2211yescharacteristic, complemented
47#19149603C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic
51#19149600C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic, complemented
54C2×Q16×C173C2211yescharacteristic, complemented
55C2×Q16×C173C2211yescharacteristic, complemented
56C2×SD16×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic
58C4⋊Q8×C173C2211yescharacteristic, complemented
59#124×C173C2211yescharacteristic
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64C2×Dic346—21no—
65C2×Dic346—21no—
66C2×Dic346C2311yescharacteristic
67Q8×D173—22no—
68Q8×D173—22no—
69Q8×D173—41no—
70Q8×D173—41no—
71C2×D692—21no—
72C2×D692D411yescharacteristic
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76D4×D173—22no—
77D4×D173—41no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83C2×C4×D173C2311yescharacteristic
84#14583816—22no—
85#14583815D414yescomplemented
86#14583814—22no—
87#14583813—22no—
88#14583813—22no—
89#14583809—22no—
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C8×D173—22no—
93C4⋊Dic173D411yescharacteristic
94C4⋊Dic173C2311yescharacteristic
95C4⋊Dic173C2311yescharacteristic
96#14583805—21no—
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100C22⋊Dic173—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803D411yescharacteristic
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C2×Q8×C173C2311yescharacteristic
109C2×Q8×C173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112Q16×C173—22no—
113Q16×C173—41no—
114SD16×C173—22no—
115SD16×C173—41no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173—21no—
119C4⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173—81no—
127C22×D173C2×D411yescharacteristic
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—42no—
131Dic346—42no—
132Dic346C2×D412yes—
133D692—22no—
134D692—21no—
135D692—42no—
136D692C2×D412yes—
137C22⋊D173—42no—
138C22⋊D173—81no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C4×D173—21no—
145C4×D173—21no—
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—42no—
149C4×D173—42no—
150C4×D173—42no—
151C4×D173—81no—
152#13063436—21no—
153#13063436—21no—
154C23×C173—21no—
155Q8×C173—21no—
156Q8×C173—21no—
157Q8×C173—21no—
158Q8×C173—21no—
159Q8×C173—21no—
160Q8×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173C2×D411yescharacteristic
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C8×C173—21no—
170C8×C173C2×D412yes—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—22no—
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C2×D173#13712yescomplemented
185C4×C173—21no—
186C4×C173—21no—
187C4×C173—41no—
188C4×C173—41no—
189C4×C173—41no—
190C4×C173C22×D411yescharacteristic
191C4×C173H(5,𝔽2)11yescharacteristic
192Dic173—21no—
193Dic173—21no—
194Dic173—22no—
195Dic173—41no—
196Dic173—41no—
197C2×C173—81no—
198C2×C173D4211yescharacteristic
199C2×C173C2×#13711yescharacteristic
200C2×C173#57711yescharacteristic
201D173—22no—
202D173—41no—
203D173—42no—
204C173#326611yescharacteristic, complemented, Sylow
205#3266—1731nomaximal, Sylow
206C2×#137—1731no—
207C22×SD16—1731no—
208D4×Q8—1731no—
209#546—1731no—
210#497D17311yescharacteristic, complemented
211#468—1731no—
212#467—1731no—
213#464—1731no—
214#461—1731no—
215#452—1731no—
216#451—1731no—
217#450—1731no—
218#448—1731no—
219#440—1731no—
220#435—1731no—
221C2×#41—1731no—
222C22×Q8—1731no—
223C22×Q8—1731no—
224C22×D4—1731no—
225#137—3464no—
226C2×Q16—1731no—
227C2×Q16C2×D17311yescharacteristic, complemented
228C2×Q16C2×D17311yescharacteristic, complemented
229C2×SD16—3464no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16C2×D17311yescharacteristic
235C2×SD16C2×D17311yescharacteristic
236C2×M16—1731no—
237C22×C8—1731no—
238C4⋊Q8—3461no—
239C4⋊Q8C2×D17311yescharacteristic, complemented
240#124C2×D17311yescharacteristic
241#123—3461no—
242C22⋊Q8—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8—1731no—
246C22⋊Q8—1731no—
247#121—3461no—
248#121—1731no—
249#120—3461no—
250C4×Q8—1731no—
251C4×D4—3461no—
252C4×D4—1731no—
253C4×D4—1731no—
254C2×C22⋊C4—3461no—
255#107—1731no—
256C4⋊C8—1731no—
257#103—1731no—
258#103—1731no—
259#103—1731no—
260#103—1731no—
261#102—1731no—
262#102—1731no—
263C22⋊C8—1731no—
264C22⋊C8—1731no—
265#97C2×D17311yescharacteristic, complemented
266C24—3461no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3461no—
270C2×Q8—3461no—
271C2×Q8—3462no—
272C2×Q8—3462no—
273C2×Q8—1731no—
274C2×Q8—6921no—
275C2×Q8—6921no—
276C2×Q8C22×D17311yescharacteristic
277C2×Q8C22×D17311yescharacteristic
278C2×Q8C22×D17311yescharacteristic
279C2×D4—3461no—
280C2×D4—3462no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—1731no—
284C2×D4—1731no—
285C2×D4—6921no—
286C2×D4C22×D17311yescharacteristic
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292C22×C4—1731no—
293Q16—22no—
294Q16—41no—
295Q16—3462no—
296SD16—22no—
297SD16—41no—
298SD16—3462no—
299SD16—3462no—
300SD16—3462no—
301SD16—1734no—
302M16—3462no—
303C2×C8—3462no—
304C2×C8—1731no—
305C2×C8—1731no—
306C2×C8C22×D17311yescharacteristic
307C2×C8C22×D17311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—3461no—
311C4⋊C4—3461no—
312C4⋊C4—3461no—
313C4⋊C4—3461no—
314C4⋊C4—1731no—
315C4⋊C4—1731no—
316C4⋊C4—1731no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—1731no—
325C22⋊C4—1731no—
326C42—3461no—
327C42C22×D17311yescharacteristic
328C23—21no—
329C23—3461no—
330C23—13841no—
331C23—1731no—
332C23—1731no—
333C23—6922no—
334C23—6922no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—3461no—
342Q8—1732no—
343Q8—6922no—
344Q8—6922no—
345D4—21no—
346D4—21no—
347D4—3462no—
348D4—3461no—
349D4—13841no—
350D4—1732no—
351D4—6922no—
352D4—6922no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—13841no—
364C2×C4—1731no—
365C2×C4—1731no—
366C2×C4—6922no—
367C2×C4—6922no—
368C2×C4—6922no—
369C2×C4D4×D17311yescharacteristic
370C2×C4D4×D17311yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—21no—
373C8—3461no—
374C8—3461no—
375C8D4×D17312yes—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—3461no—
380C22—3461no—
381C22—3461no—
382C22—3461no—
383C22—3461no—
384C22—3462no—
385C22—13844no—
386C22—1732no—
387C22—6921no—
388C22—6921no—
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3461no—
396C4—3461no—
397C4—3462no—
398C4—6921no—
399C4—6921no—
400C4C2×D4×D17311yescharacteristic
401C4#1914972411yescharacteristic
402C2—81no—
403C2—3462no—
404C2—6921no—
405C2—6922no—
406C2#2601212611yescharacteristic
407C2#137×D17311yescharacteristic
408C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#497×C173C2
socleC22×C173C22×D4
prime radical#26012134C2
hypercenter#497D173
hypocenterC173#3266
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×SD16×D173C2semidirect
3. #26012156C2semidirect
5. #26011734C2semidirect
6. #26011729C2semidirect
7. #26011710C2semidirect
8. #26011693C2semidirect
9. #26011652C2semidirect
10. #26011645C2semidirect
11. #26011640C2semidirect
12. #26011628C2semidirect
13. #26011593C2semidirect
14. #26011564C2semidirect
15. #497×C173C2semidirect
18. C2×Q8×D173C22semidirect
21. C2×#14583816C22semidirect
27. C2×#14583809C22semidirect
35. #19149655C22semidirect
39. C4×Dic346C22semidirect
44. #19149613C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149604C22semidirect
47. #19149603C22semidirect
48. #19149603C22semidirect
49. #19149603C22semidirect
53. #19149591C22semidirect
54. C2×Q16×C173C22semidirect
55. C2×Q16×C173C22semidirect
58. C4⋊Q8×C173C22semidirect
60. #97×C173C22semidirect
85. #14583815D4semidirect
184. C2×D173#137semidirect
204. C173#3266semidirect
210. #497D173semidirect
227. C2×Q16C2×D173semidirect
228. C2×Q16C2×D173semidirect
239. C4⋊Q8C2×D173semidirect
265. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×SD16×D1732
#260121562
#260121342
#260117342
#260117292
#260117102
#260116932
#260116522
#260116452
#260116402
#260116282
#260115932
#260115642
#497×C1732
#3266173