Finite Group Databasebeta

← #36365080

Subgroups — #36365080

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×#19149712C211yesmaximal, characteristic, complemented
2C2×SD16×D173C211yesmaximal, characteristic, complemented
3#26012154C211yesmaximal, characteristic, complemented
4#26012138C211yesmaximal, characteristic
5#26011882C211yesmaximal, characteristic, complemented
6#26011733C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011695C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011648C211yesmaximal, characteristic, complemented
11#26011643C211yesmaximal, characteristic, complemented
12#26011629C211yesmaximal, characteristic, complemented
13#26011590C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic, complemented
18C2×Q8×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic
20#19149712—24no—
21C2×Dic692C2211yescharacteristic, complemented
22C2×Dic692C2211yescharacteristic, complemented
23C2×#14583816C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic
25C2×#14583815C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583814C2211yescharacteristic
28C2×#14583813C2211yescharacteristic
29SD16×D173—24no—
30C2×C8×D173C2211yescharacteristic
31#19149678—21no—
32#19149677C2211yescharacteristic, complemented
33#19149666C2211yescharacteristic
34#19149661—21no—
35#19149659C2211yescharacteristic
36#19149659C2211yescharacteristic
37#19149657—21no—
38#19149656—21no—
39#19149650—21no—
40#19149647—21no—
41#19149644C2211yescharacteristic, complemented
42#19149641C2211yescharacteristic
43C4×C22⋊D173C2211yescharacteristic
44#19149639—21no—
45C22⋊C4×D173—21no—
46#19149608C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149602C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic
51C22⋊#13063436C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic
53#19149589C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic
55C2×M16×C173C2211yescharacteristic, complemented
56C22⋊Q8×C173C2211yescharacteristic, complemented
57#121×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic
61C23×D173—21no—
62#14583826—22no—
63#14583826—41no—
64C2×Dic346—21no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70C2×Dic346C2311yescharacteristic
71Q8×D173—22no—
72Q8×D173—41no—
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173D411yescharacteristic
77D4×D173—22no—
78D4×D173—41no—
79C22×Dic173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173C2311yescharacteristic
84Dic692—22no—
85Dic692—22no—
86#14583816—41no—
87#14583815—22no—
88#14583815D414yescomplemented
89#14583814—22no—
90#14583814—41no—
91#14583813—22no—
92C2×#13063436C2311yescharacteristic
93C2×#13063436C2311yescharacteristic
94C8×D173—22no—
95C4⋊Dic173—21no—
96C4⋊Dic173—21no—
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803D411yescharacteristic
108C4×Dic173—21no—
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114SD16×C173—22no—
115M16×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173—21no—
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173—81no—
127C22×D173C2×D411yescharacteristic
128Dic346—21no—
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—21no—
133Dic346—42no—
134Dic346—42no—
135Dic346C2×D412yes—
136D692—21no—
137D692C2×D412yes—
138C22⋊D173—22no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—42no—
145C2×Dic173C2×D411yescharacteristic
146C2×Dic173C2×D411yescharacteristic
147C2×Dic173C2×D411yescharacteristic
148C4×D173—21no—
149C4×D173—21no—
150C4×D173—42no—
151C4×D173—42no—
152C4×D173—81no—
153#13063436—21no—
154#13063436—21no—
155C23×C173—21no—
156C23×C173C2×D411yescharacteristic
157Q8×C173—21no—
158Q8×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—81no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C8×C173—21no—
170C8×C173C2×D412yes—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173C22×D411yescharacteristic
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—22no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C2×D173#13712yescomplemented
185C4×C173—21no—
186C4×C173—41no—
187C4×C173—41no—
188C4×C173—41no—
189C4×C173C22×D411yescharacteristic
190C4×C173H(5,𝔽2)11yescharacteristic
191Dic173—21no—
192Dic173—22no—
193Dic173—21no—
194Dic173—21no—
195Dic173—41no—
196Dic173—41no—
197C2×C173—41no—
198C2×C173—81no—
199C2×C173D4211yescharacteristic
200C2×C173C2×#13711yescharacteristic
201C2×C173#57711yescharacteristic
202D173—22no—
203D173—42no—
204C173#326611yescharacteristic, complemented, Sylow
205#3266—1731nomaximal, Sylow
206C2×#137—1731no—
207C22×SD16—1731no—
208D4×Q8—1731no—
209#546—1731no—
210#497—1731no—
211#468D17311yescharacteristic, complemented
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×Q16—1731no—
228C2×Q16—1731no—
229C2×SD16—3464no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16C2×D17311yescharacteristic
236C2×M16C2×D17311yescharacteristic, complemented
237C22×C8—1731no—
238C4⋊Q8—3461no—
239C4⋊Q8—1731no—
240#124—1731no—
241#123—3461no—
242C22⋊Q8—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8—1731no—
246C22⋊Q8C2×D17311yescharacteristic, complemented
247#121—3461no—
248#121C2×D17311yescharacteristic
249#120—3461no—
250C4×Q8—1731no—
251C4×D4—3461no—
252C4×D4—1731no—
253C4×D4—1731no—
254C2×C22⋊C4—3461no—
255#107C2×D17311yescharacteristic, complemented
256C4⋊C8—1731no—
257#103—1731no—
258#103—1731no—
259#103—1731no—
260#103C2×D17311yescharacteristic, complemented
261#102—1731no—
262#102C2×D17311yescharacteristic
263C22⋊C8—1731no—
264C22⋊C8—1731no—
265#97—1731no—
266C24—3461no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3461no—
270C2×Q8—3462no—
271C2×Q8—3461no—
272C2×Q8—3462no—
273C2×Q8—1731no—
274C2×Q8—1731no—
275C2×Q8—1731no—
276C2×Q8—6921no—
277C2×Q8—6921no—
278C2×Q8C22×D17311yescharacteristic
279C2×D4—21no—
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×C4C22×D17311yescharacteristic
293Q16—3462no—
294Q16—3462no—
295Q16—6921no—
296SD16—22no—
297SD16—3462no—
298SD16—3462no—
299SD16—3462no—
300SD16—1734no—
301SD16—6921no—
302M16—22no—
303C2×C8—3462no—
304C2×C8—1731no—
305C2×C8—1731no—
306C2×C8C22×D17311yescharacteristic
307C2×C8C22×D17311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3461no—
310C4⋊C4—3461no—
311C4⋊C4—3461no—
312C4⋊C4—3461no—
313C4⋊C4—3461no—
314C4⋊C4—1731no—
315C4⋊C4C22×D17311yescharacteristic
316C4⋊C4C22×D17311yescharacteristic
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—
327C42—1731no—
328C23—21no—
329C23—3461no—
330C23—13841no—
331C23—1731no—
332C23—6922no—
333C23—6922no—
334C23D4×D17311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3461no—
338Q8—3461no—
339Q8—3461no—
340Q8—3461no—
341Q8—3461no—
342Q8—1732no—
343Q8—6922no—
344Q8—6922no—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—3462no—
349D4—3461no—
350D4—1732no—
351D4—6922no—
352D4—6922no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
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—1731no—
367C2×C4—6922no—
368C2×C4—6922no—
369C2×C4—6922no—
370C2×C4D4×D17311yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—21no—
373C8—3461no—
374C8—3461no—
375C8D4×D17312yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—3461no—
383C22—3461no—
384C22—3462no—
385C22—13844no—
386C22—1732no—
387C22—6921no—
388C22—6921no—
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—3461no—
395C4—3461no—
396C4—3461no—
397C4—3462no—
398C4—6921no—
399C4—6921no—
400C4C2×D4×D17311yescharacteristic
401C4#1914972511yescharacteristic
402C2—41no—
403C2—81no—
404C2—3462no—
405C2—6922no—
406C2#2601212511yescharacteristic
407C2#137×D17311yescharacteristic
408C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#26012138C2
hypercenter#468D173
hypocenterC173#3266
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C2×SD16×D173C2semidirect
3. #26012154C2semidirect
5. #26011882C2semidirect
6. #26011733C2semidirect
7. #26011712C2semidirect
8. #26011695C2semidirect
9. #26011651C2semidirect
10. #26011648C2semidirect
11. #26011643C2semidirect
12. #26011629C2semidirect
13. #26011590C2semidirect
14. #26011560C2semidirect
15. #468×C173C2semidirect
17. C22×Dic346C22semidirect
21. C2×Dic692C22semidirect
22. C2×Dic692C22semidirect
23. C2×#14583816C22semidirect
32. #19149677C22semidirect
41. #19149644C22semidirect
46. #19149608C22semidirect
47. #19149602C22semidirect
48. #19149602C22semidirect
49. #19149602C22semidirect
51. C22⋊#13063436C22semidirect
53. #19149589C22semidirect
55. C2×M16×C173C22semidirect
56. C22⋊Q8×C173C22semidirect
58. #107×C173C22semidirect
59. #103×C173C22semidirect
88. #14583815D4semidirect
184. C2×D173#137semidirect
204. C173#3266semidirect
211. #468D173semidirect
236. C2×M16C2×D173semidirect
246. C22⋊Q8C2×D173semidirect
255. #107C2×D173semidirect
260. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×SD16×D1732
#260121542
#260121382
#260118822
#260117332
#260117122
#260116952
#260116512
#260116482
#260116432
#260116292
#260115902
#260115602
#468×C1732
#3266173