Finite Group Databasebeta

← #36365134

Subgroups — #36365134

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×#19149710C211yesmaximal, characteristic, complemented
2C22×#14583816C211yesmaximal, characteristic, complemented
3#26012150C211yesmaximal, characteristic
4#26011883C211yesmaximal, characteristic, complemented
5#26011747C211yesmaximal, characteristic, complemented
6#26011722C211yesmaximal, characteristic, complemented
7#26011713C211yesmaximal, characteristic, complemented
8#26011698C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011647C211yesmaximal, characteristic, complemented
11#26011646C211yesmaximal, characteristic, complemented
12#26011630C211yesmaximal, characteristic, complemented
13#26011596C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15D4×Q8×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18#19149710—24no—
19C2×#14583816—24no—
20C2×#14583816C2211yescharacteristic
21C2×#14583816C2211yescharacteristic
22C2×#14583816C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25C2×#14583816C2211yescharacteristic
26C2×#14583813C2211yescharacteristic
27C2×#14583813C2211yescharacteristic
28C2×#14583813C2211yescharacteristic
29C2×#14583810C2211yescharacteristic
30C22×#13063436C2211yescharacteristic
31#19149667C2211yescharacteristic
32#19149665—21no—
33#19149658C2211yescharacteristic
34#19149657—21no—
35#19149653C2211yescharacteristic
36#19149651—21no—
37C4×Dic346C2211yescharacteristic
38#19149639—21no—
39#19149610C2211yescharacteristic
40C4⋊#13063436C2211yescharacteristic
41#19149603C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149599C2211yescharacteristic
47C22⋊#13063436C2211yescharacteristic
48C22⋊#13063436C2211yescharacteristic
49#19149590C2211yescharacteristic
50C22×Q8×C173C2211yescharacteristic
51C22×Q8×C173C2211yescharacteristic
52C4⋊Q8×C173—21no—
53C4⋊Q8×C173C2211yescharacteristic
54C22⋊Q8×C173—21no—
55C22⋊Q8×C173—21no—
56C22⋊Q8×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58C4×Q8×C173C2211yescharacteristic
59C4×D4×C173—21no—
60C4×D4×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—41no—
63C2×Dic346—22no—
64C2×Dic346—21no—
65C2×Dic346—41no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73#14583816—22no—
74#14583816—22no—
75#14583816—22no—
76#14583816—22no—
77#14583816—41no—
78#14583816D414yescomplemented
79#14583813—22no—
80#14583813—22no—
81#14583813—41no—
82#14583810—22no—
83C2×#13063436—22no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C2×Q8×C173—21no—
97C2×Q8×C173—21no—
98C2×Q8×C173—22no—
99C2×Q8×C173—22no—
100C2×Q8×C173—41no—
101C2×Q8×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×Q8×C173C2311yescharacteristic
105C2×D4×C173D411yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173—21no—
108C22×C4×C173C2311yescharacteristic
109C22×C4×C173C2311yescharacteristic
110C4⋊C4×C173—21no—
111C4⋊C4×C173—21no—
112C4⋊C4×C173—21no—
113C4⋊C4×C173—21no—
114C4⋊C4×C173—21no—
115C4⋊C4×C173D411yescharacteristic
116C4⋊C4×C173C2311yescharacteristic
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173D411yescharacteristic
120C22⋊C4×C173D411yescharacteristic
121C42×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—42no—
128Dic346C2×D412yes—
129D692—21no—
130D692—21no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C4×D173—42no—
138#13063436—21no—
139#13063436—21no—
140#13063436—21no—
141#13063436C2×D412yes—
142C23×C173C2×D411yescharacteristic
143C23×C173C2×D411yescharacteristic
144Q8×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148Q8×C173—21no—
149Q8×C173—42no—
150Q8×C173—42no—
151Q8×C173C2×D412yes—
152D4×C173—22no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—42no—
162C2×C4×C173—42no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2×D411yescharacteristic
166C2×C4×C173C2×D411yescharacteristic
167C2×C4×C173C2411yescharacteristic
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173#13712yescomplemented
174C22×C173C22×D411yescharacteristic
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—22no—
181C4×C173—21no—
182C4×C173—21no—
183C4×C173—41no—
184C4×C173—41no—
185C4×C173—41no—
186C4×C173C22×D411yescharacteristic
187C4×C173H(5,𝔽2)11yescharacteristic
188Dic173—22no—
189Dic173—41no—
190Dic173—41no—
191C2×C173—22no—
192C2×C173—41no—
193C2×C173D4211yescharacteristic
194C2×C173C2×#13711yescharacteristic
195C2×C173#57811yescharacteristic
196D173—81no—
197C173#327011yescharacteristic, complemented, Sylow
198#3270—1731nomaximal, Sylow
199C2×#137—1731no—
200C22×Q16—1731no—
201D4×Q8D17311yescharacteristic, complemented
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#441—1731no—
213#435—1731no—
214C2×#41—1731no—
215C22×Q8—1731no—
216C22×Q8C2×D17311yescharacteristic
217C22×Q8C2×D17311yescharacteristic
218#137—3464no—
219C2×Q16—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×Q16—1731no—
223C2×Q16—1731no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×M16—1731no—
230C22×C8—1731no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D17311yescharacteristic
233#124—3461no—
234#124—1731no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—3461no—
238C22⋊Q8—1731no—
239C22⋊Q8C2×D17311yescharacteristic
240C22⋊Q8C2×D17311yescharacteristic
241#121—3461no—
242#121—1731no—
243C4×Q8—1731no—
244C4×Q8C2×D17311yescharacteristic
245C4×D4—21no—
246C4×D4—3461no—
247C4×D4C2×D17311yescharacteristic
248#106—1731no—
249C4⋊C8—1731no—
250#103—1731no—
251#103—1731no—
252#103—1731no—
253#103—1731no—
254#103—1731no—
255#102—1731no—
256C22⋊C8—1731no—
257C22⋊C8—1731no—
258#97—1731no—
259#41—3462no—
260#41—6921no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—3462no—
268C2×Q8—3461no—
269C2×Q8—1731no—
270C2×Q8—1731no—
271C2×Q8—6921no—
272C2×Q8C22×D17311yescharacteristic
273C2×Q8C22×D17311yescharacteristic
274C2×Q8C22×D17311yescharacteristic
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—1731no—
278C2×D4C22⋊D17311yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4C22×D17311yescharacteristic
284C22×C4C22×D17311yescharacteristic
285Q16—3462no—
286Q16—3462no—
287Q16—3462no—
288Q16—3462no—
289Q16—1734no—
290Q16—6921no—
291SD16—3462no—
292SD16—3462no—
293SD16—6921no—
294M16—3462no—
295C2×C8—3462no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—1731no—
299C2×C8—1731no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—1731no—
308C4⋊C4C22⋊D17311yescharacteristic
309C4⋊C4C22×D17311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4C22⋊D17311yescharacteristic
317C22⋊C4C22⋊D17311yescharacteristic
318C42—21no—
319C42—3461no—
320C42C22×D17311yescharacteristic
321C23—3461no—
322C23C2×C22⋊D17311yescharacteristic
323C23C2×C22⋊D17311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—3461no—
332Q8—3461no—
333Q8—3461no—
334Q8—1732no—
335Q8—6922no—
336Q8D4×D17312yes—
337D4—22no—
338D4—3461no—
339D4—3461no—
340D4—6922no—
341D4—6922no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—6922no—
356C2×C4—6922no—
357C2×C4C2×C22⋊D17311yescharacteristic
358C2×C4C2×C22⋊D17311yescharacteristic
359C2×C4C2×C22⋊D17311yescharacteristic
360C2×C4C2×C22⋊D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—3461no—
363C8—3461no—
364C8—3461no—
365C8—1732no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22#1914971012yescomplemented
375C22C22×C22⋊D17311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3462no—
385C4—6921no—
386C4—6921no—
387C4C2×D4×D17311yescharacteristic
388C4#1914972411yescharacteristic
389C2—22no—
390C2—41no—
391C2—13841no—
392C2D4×C22⋊D17311yescharacteristic
393C2C2×#1914971011yescharacteristic
394C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingD4×Q8×C173C2
socleC22×C173C22×D4
prime radical#26012150C2
hypercenterD4×Q8D173
hypocenterC173#3270
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19149710C2semidirect
2. C22×#14583816C2semidirect
4. #26011883C2semidirect
5. #26011747C2semidirect
6. #26011722C2semidirect
7. #26011713C2semidirect
8. #26011698C2semidirect
9. #26011651C2semidirect
10. #26011647C2semidirect
11. #26011646C2semidirect
12. #26011630C2semidirect
13. #26011596C2semidirect
14. #26011563C2semidirect
15. D4×Q8×C173C2semidirect
78. #14583816D4semidirect
173. C22×C173#137semidirect
197. C173#3270semidirect
201. D4×Q8D173semidirect
374. C22#19149710semidirect

Maximal subgroups

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

subgroupindex
C2×#191497102
C22×#145838162
#260121502
#260118832
#260117472
#260117222
#260117132
#260116982
#260116512
#260116472
#260116462
#260116302
#260115962
#260115632
D4×Q8×C1732
#3270173