Finite Group Databasebeta

← #36365135

Subgroups — #36365135

Order 22144.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C2×Q16×D173C211yesmaximal, characteristic, complemented
3Q8×D692C211yesmaximal, characteristic, complemented
4#26012153C211yesmaximal, characteristic
5#26011885C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011713C211yesmaximal, characteristic, complemented
9#26011701C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011648C211yesmaximal, characteristic, complemented
12#26011648C211yesmaximal, characteristic, complemented
13#26011631C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#441×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×Q8×D173C2211yescharacteristic
20#19149711—24no—
21C2×Dic692C2211yescharacteristic
22C2×Dic692C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25C2×#14583816C2211yescharacteristic
26Q16×D173—24no—
27C2×#14583815C2211yescharacteristic
28C2×#14583815C2211yescharacteristic
29C2×#14583813C2211yescharacteristic
30C2×#14583809C2211yescharacteristic
31C2×C8×D173C2211yescharacteristic
32#19149679—21no—
33#19149679C2211yescharacteristic
34#19149667—21no—
35#19149667C2211yescharacteristic
36#19149659—21no—
37#19149659—21no—
38#19149659—21no—
39#19149659C2211yescharacteristic
40#19149659C2211yescharacteristic
41#19149659C2211yescharacteristic
42#19149654—21no—
43#19149654C2211yescharacteristic
44C4×D692—21no—
45C4×D692—21no—
46C4×D692C2211yescharacteristic
47C4⋊#13063436C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149599C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C4×Q8×C173C2211yescharacteristic
56C4×Q8×C173C2211yescharacteristic
57#106×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#97×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—41no—
63C2×Dic346—21no—
64C2×Dic346—21no—
65C2×Dic346—21no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×Dic346C2311yescharacteristic
69Q8×D173—22no—
70Q8×D173—22no—
71Q8×D173—22no—
72Q8×D173—41no—
73Q8×D173—41no—
74Q8×D173—41no—
75C2×D692—21no—
76C2×D692—21no—
77C2×D692D411yescharacteristic
78C2×D692C2311yescharacteristic
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173C2311yescharacteristic
84C2×C4×D173C2311yescharacteristic
85Dic692—22no—
86Dic692—41no—
87#14583816—22no—
88#14583816—22no—
89#14583816—22no—
90#14583815—22no—
91#14583815—41no—
92#14583813—22no—
93#14583809—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C8×D173—22no—
97C4⋊Dic173—21no—
98C4⋊Dic173—21no—
99C4⋊Dic173—21no—
100C4⋊Dic173—21no—
101C4⋊Dic173—21no—
102C4⋊Dic173D411yescharacteristic
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109#14583803D411yescharacteristic
110#14583803D411yescharacteristic
111C2×Q8×C173C2311yescharacteristic
112C2×Q8×C173C2311yescharacteristic
113Q16×C173D414yescomplemented
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173—21no—
117C4⋊C4×C173—21no—
118C4⋊C4×C173C2311yescharacteristic
119C4⋊C4×C173C2311yescharacteristic
120C42×C173—21no—
121C42×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125C22×D173C2×D411yescharacteristic
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—42no—
133Dic346—42no—
134Dic346—42no—
135D692—22no—
136D692—21no—
137D692—21no—
138D692—42no—
139D692—42no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C4×D173—21no—
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—21no—
149C4×D173—42no—
150C4×D173—42no—
151C4×D173—42no—
152C4×D173—42no—
153#13063436—21no—
154#13063436—21no—
155Q8×C173—21no—
156Q8×C173—21no—
157Q8×C173C2×D412yes—
158Q8×C173C2×D412yes—
159C2×C4×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×C173C2×D412yes—
168C22×C173C22×D411yescharacteristic
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173#13712yescomplemented
178C4×C173—22no—
179C4×C173—22no—
180C4×C173—21no—
181C4×C173—21no—
182C4×C173—41no—
183C4×C173—41no—
184C4×C173C22×D411yescharacteristic
185C4×C173H(5,𝔽2)11yescharacteristic
186Dic173—21no—
187Dic173—21no—
188Dic173—41no—
189Dic173—41no—
190Dic173—41no—
191C2×C173D4211yescharacteristic
192C2×C173C2×#13711yescharacteristic
193C2×C173#57811yescharacteristic
194D173—22no—
195D173—41no—
196D173—81no—
197C173#327011yescharacteristic, complemented, Sylow
198#3270—1731nomaximal, Sylow
199C2×#137—1731no—
200C22×Q16—1731no—
201D4×Q8—1731no—
202#548—1731no—
203#497—1731no—
204#470—1731no—
205#465—1731no—
206#464—1731no—
207#462—1731no—
208#452—1731no—
209#451—1731no—
210#451—1731no—
211#448—1731no—
212#441D17311yescharacteristic, complemented
213#435—1731no—
214C2×#41—1731no—
215C22×Q8—1731no—
216C22×Q8—1731no—
217C22×Q8—1731no—
218#137—3464no—
219C2×Q16—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×Q16—1731no—
223C2×Q16—1731no—
224C2×Q16—1731no—
225C2×Q16C2×D17311yescharacteristic
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×M16—1731no—
230C22×C8—1731no—
231C4⋊Q8—3461no—
232C4⋊Q8—1731no—
233#124—3461no—
234#124—1731no—
235C22⋊Q8—3461no—
236C22⋊Q8—3461no—
237C22⋊Q8—3461no—
238C22⋊Q8—1731no—
239C22⋊Q8—1731no—
240C22⋊Q8—1731no—
241#121—3461no—
242#121—1731no—
243C4×Q8C2×D17311yescharacteristic
244C4×Q8C2×D17311yescharacteristic
245C4×D4—3461no—
246C4×D4—3461no—
247C4×D4—1731no—
248#106C2×D17311yescharacteristic
249C4⋊C8—1731no—
250#103—1731no—
251#103—1731no—
252#103—1731no—
253#103C2×D17311yescharacteristic
254#103C2×D17311yescharacteristic
255#102—1731no—
256C22⋊C8—1731no—
257C22⋊C8—1731no—
258#97C2×D17311yescharacteristic
259#41—3462no—
260#41—6921no—
261C2×Q8—3461no—
262C2×Q8—3461no—
263C2×Q8—3461no—
264C2×Q8—3462no—
265C2×Q8—3462no—
266C2×Q8—3462no—
267C2×Q8—1731no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×Q8—6921no—
271C2×Q8—6921no—
272C2×Q8—6921no—
273C2×Q8C22×D17311yescharacteristic
274C2×Q8C22×D17311yescharacteristic
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—1731no—
278C2×D4—1731no—
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4—1731no—
284C22×C4—1731no—
285Q16—3462no—
286Q16—3462no—
287Q16—3462no—
288Q16—3462no—
289Q16—6921no—
290Q16D69214yescomplemented
291SD16—3462no—
292SD16—3462no—
293SD16—6921no—
294M16—3462no—
295C2×C8—3462no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8C22×D17311yescharacteristic
299C2×C8C22×D17311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—3461no—
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—1731no—
308C4⋊C4C22×D17311yescharacteristic
309C4⋊C4C22×D17311yescharacteristic
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—1731no—
317C22⋊C4—1731no—
318C42—21no—
319C42—21no—
320C42C22×D17311yescharacteristic
321C23—3461no—
322C23—1731no—
323C23—1731no—
324Q8—21no—
325Q8—21no—
326Q8—3461no—
327Q8—3461no—
328Q8—3461no—
329Q8—3461no—
330Q8—3461no—
331Q8—3461no—
332Q8—6922no—
333Q8—6922no—
334Q8—6922no—
335Q8C2×D69212yes—
336Q8C2×D69212yes—
337D4—3462no—
338D4—3461no—
339D4—3461no—
340D4—6922no—
341D4—6922no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—1731no—
354C2×C4—1731no—
355C2×C4—6922no—
356C2×C4—6922no—
357C2×C4—6922no—
358C2×C4—6922no—
359C2×C4D4×D17311yescharacteristic
360C2×C4D4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—3461no—
364C8—3461no—
365C8C2×D69212yes—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—3461no—
370C22—3461no—
371C22—1732no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—3461no—
383C4—3461no—
384C4—6921no—
385C4—6921no—
386C4—6921no—
387C4C22×D69211yescharacteristic
388C4#1914972511yescharacteristic
389C2—3462no—
390C2—13841no—
391C2—6921no—
392C2D4×D69211yescharacteristic
393C2#137×D17311yescharacteristic
394C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#441×C173C2
socleC22×C173C22×D4
prime radical#26012153C2
hypercenter#441D173
hypocenterC173#3270
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×Q16×D173C2semidirect
3. Q8×D692C2semidirect
5. #26011885C2semidirect
6. #26011750C2semidirect
7. #26011721C2semidirect
8. #26011713C2semidirect
9. #26011701C2semidirect
10. #26011656C2semidirect
11. #26011648C2semidirect
12. #26011648C2semidirect
13. #26011631C2semidirect
14. #26011564C2semidirect
15. #441×C173C2semidirect
113. Q16×C173D4semidirect
177. C2×D173#137semidirect
197. C173#3270semidirect
212. #441D173semidirect
290. Q16D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×Q16×D1732
Q8×D6922
#260121532
#260118852
#260117502
#260117212
#260117132
#260117012
#260116562
#260116482
#260116312
#260115642
#441×C1732
#3270173