Finite Group Databasebeta

← #36365074

Subgroups — #36365074

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×#19149710C211yesmaximal, characteristic, complemented
2C22×#14583813C211yesmaximal, characteristic, complemented
3#26012137C211yesmaximal, characteristic
4#26011883C211yesmaximal, characteristic, complemented
5#26011739C211yesmaximal, characteristic, complemented
6#26011730C211yesmaximal, characteristic, complemented
7#26011713C211yesmaximal, characteristic, complemented
8#26011693C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011647C211yesmaximal, characteristic, complemented
11#26011641C211yesmaximal, characteristic, complemented
12#26011630C211yesmaximal, characteristic, complemented
13#26011589C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15D4×Q8×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18#19149710—24no—
19C2×#14583816C2211yescharacteristic, complemented
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583813—24no—
23C2×#14583813C2211yescharacteristic
24C2×#14583813C2211yescharacteristic
25C2×#14583813C2211yescharacteristic
26C2×#14583813C2211yescharacteristic
27C2×#14583813C2211yescharacteristic
28C2×#14583813C2211yescharacteristic
29C2×#14583810C2211yescharacteristic, complemented
30C22×#13063436C2211yescharacteristic
31#19149667C2211yescharacteristic
32#19149664—21no—
33#19149658C2211yescharacteristic
34#19149653C2211yescharacteristic
35#19149652—21no—
36#19149647—21no—
37C4×D692C2211yescharacteristic
38C22⋊C4×D173—21no—
39#19149612C2211yescharacteristic, complemented
40C4⋊#13063436C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47C22⋊#13063436C2211yescharacteristic
48C22⋊#13063436C2211yescharacteristic, complemented
49#19149590C2211yescharacteristic, complemented
50C22×Q8×C173C2211yescharacteristic
51C22×Q8×C173C2211yescharacteristic, complemented
52C4⋊Q8×C173—21no—
53C4⋊Q8×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173—21no—
55C22⋊Q8×C173—21no—
56C22⋊Q8×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic, complemented
58C4×Q8×C173C2211yescharacteristic, complemented
59C4×D4×C173—21no—
60C4×D4×C173C2211yescharacteristic
61C23×D173—21no—
62#14583826—22no—
63#14583826—41no—
64C2×Dic346C2311yescharacteristic
65C2×D692—22no—
66C2×D692—21no—
67C2×D692—41no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74#14583816—22no—
75#14583816—22no—
76#14583816—41no—
77#14583813—22no—
78#14583813—22no—
79#14583813—22no—
80#14583813—22no—
81#14583813—41no—
82#14583813D414yescomplemented
83#14583810—22no—
84C2×#13063436—22no—
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C2×#13063436C2311yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—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—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128Dic346—21no—
129Dic346—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—42no—
134D692C2×D412yes—
135C22⋊D173—42no—
136C22⋊D173—81no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C4×D173—42no—
140C4×D173—81no—
141#13063436—21no—
142#13063436—21no—
143#13063436—21no—
144#13063436C2×D412yes—
145C23×C173C2×D411yescharacteristic
146C23×C173C2×D411yescharacteristic
147Q8×C173—21no—
148Q8×C173—21no—
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—42no—
153Q8×C173—42no—
154Q8×C173C2×D412yes—
155D4×C173—22no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—42no—
165C2×C4×C173—42no—
166C2×C4×C173C2×D411yescharacteristic
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2×D411yescharacteristic
170C2×C4×C173C2411yescharacteristic
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—21no—
176C22×C173#13712yescomplemented
177C22×C173C22×D411yescharacteristic
178C2×D173—22no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—41no—
184C2×D173—84no—
185C4×C173—21no—
186C4×C173—21no—
187C4×C173—22no—
188C4×C173—21no—
189C4×C173—21no—
190C4×C173—41no—
191C4×C173—41no—
192C4×C173—41no—
193C4×C173C22×D411yescharacteristic
194C4×C173H(5,𝔽2)11yescharacteristic
195Dic173—41no—
196Dic173—41no—
197C2×C173—22no—
198C2×C173—41no—
199C2×C173D4211yescharacteristic
200C2×C173C2×#13711yescharacteristic
201C2×C173#57711yescharacteristic
202D173—42no—
203D173—81no—
204C173#326611yescharacteristic, complemented, Sylow
205#3266—1731nomaximal, Sylow
206C2×#137—1731no—
207C22×SD16—1731no—
208D4×Q8D17311yescharacteristic, complemented
209#546—1731no—
210#497—1731no—
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×Q8C2×D17311yescharacteristic
223C22×Q8C2×D17311yescharacteristic, complemented
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×SD16—1731no—
236C2×M16—1731no—
237C22×C8—1731no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D17311yescharacteristic, complemented
240#124—1731no—
241#123—3461no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1731no—
245C22⋊Q8C2×D17311yescharacteristic
246C22⋊Q8C2×D17311yescharacteristic, complemented
247#121—3461no—
248#121—1731no—
249#120—3461no—
250C4×Q8C2×D17311yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1731no—
253C4×D4C2×D17311yescharacteristic
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#97—1731no—
266C24—3461no—
267#41—3462no—
268#41—6921no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1731no—
276C2×Q8C22×D17311yescharacteristic
277C2×Q8C22×D17311yescharacteristic
278C2×Q8C22×D17311yescharacteristic
279C2×D4—3462no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—1731no—
284C2×D4—1731no—
285C2×D4—6921no—
286C2×D4C22⋊D17311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4C22×D17311yescharacteristic
292C22×C4C22×D17311yescharacteristic
293Q16—3462no—
294Q16—3462no—
295Q16—6921no—
296SD16—3462no—
297SD16—3462no—
298SD16—3462no—
299SD16—3462no—
300SD16—1734no—
301SD16—6921no—
302M16—3462no—
303C2×C8—3462no—
304C2×C8—1731no—
305C2×C8—1731no—
306C2×C8—1731no—
307C2×C8—1731no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
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⋊C4C22⋊D17311yescharacteristic
325C22⋊C4C22⋊D17311yescharacteristic
326C42—21no—
327C42C22×D17311yescharacteristic
328C23—3461no—
329C23—3461no—
330C23—13841no—
331C23—6922no—
332C23—6922no—
333C23C2×C22⋊D17311yescharacteristic
334C23C2×C22⋊D17311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—3461no—
343Q8—3461no—
344Q8D4×D17312yes—
345D4—22no—
346D4—3461no—
347D4—3461no—
348D4—3461no—
349D4—13841no—
350D4—1732no—
351D4—6922no—
352D4—6922no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—13841no—
366C2×C4—6922no—
367C2×C4C2×C22⋊D17311yescharacteristic
368C2×C4C2×C22⋊D17311yescharacteristic
369C2×C4C2×C22⋊D17311yescharacteristic
370C2×C4C2×C22⋊D17311yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—3461no—
373C8—3461no—
374C8—3461no—
375C8—1732no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3462no—
382C22—13844no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22—6921no—
388C22#1914971012yescomplemented
389C22C22×C22⋊D17311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—6921no—
399C4—6921no—
400C4C2×D4×D17311yescharacteristic
401C4#1914972411yescharacteristic
402C2—22no—
403C2—41no—
404C2—13841no—
405C2—6922no—
406C2D4×C22⋊D17311yescharacteristic
407C2C2×#1914971011yescharacteristic
408C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingD4×Q8×C173C2
socleC22×C173C22×D4
prime radical#26012137C2
hypercenterD4×Q8D173
hypocenterC173#3266
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×#14583813C2semidirect
4. #26011883C2semidirect
5. #26011739C2semidirect
6. #26011730C2semidirect
7. #26011713C2semidirect
8. #26011693C2semidirect
9. #26011651C2semidirect
10. #26011647C2semidirect
11. #26011641C2semidirect
12. #26011630C2semidirect
13. #26011589C2semidirect
14. #26011563C2semidirect
15. D4×Q8×C173C2semidirect
19. C2×#14583816C22semidirect
20. C2×#14583816C22semidirect
21. C2×#14583816C22semidirect
29. C2×#14583810C22semidirect
39. #19149612C22semidirect
40. C4⋊#13063436C22semidirect
41. #19149603C22semidirect
42. #19149603C22semidirect
43. #19149603C22semidirect
44. #19149602C22semidirect
48. C22⋊#13063436C22semidirect
49. #19149590C22semidirect
51. C22×Q8×C173C22semidirect
53. C4⋊Q8×C173C22semidirect
57. C22⋊Q8×C173C22semidirect
58. C4×Q8×C173C22semidirect
82. #14583813D4semidirect
176. C22×C173#137semidirect
204. C173#3266semidirect
208. D4×Q8D173semidirect
223. C22×Q8C2×D173semidirect
239. C4⋊Q8C2×D173semidirect
246. C22⋊Q8C2×D173semidirect
250. C4×Q8C2×D173semidirect
388. 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×#145838132
#260121372
#260118832
#260117392
#260117302
#260117132
#260116932
#260116512
#260116472
#260116412
#260116302
#260115892
#260115632
D4×Q8×C1732
#3266173