Finite Group Databasebeta

← #36365084

Subgroups — #36365084

Order 22144.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149705C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3#26012146C211yesmaximal, characteristic, complemented
4#26011886C211yesmaximal, characteristic, complemented
5#26011747C211yesmaximal, characteristic, complemented
6#26011718C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011696C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011634C211yesmaximal, characteristic, complemented
13#26011601C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15#546×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic, complemented
18#19149705—24no—
19#19149699—24no—
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583814C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583813C2211yescharacteristic, complemented
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583810C2211yescharacteristic
30C22×#13063436C2211yescharacteristic, complemented
31#19149679C2211yescharacteristic, complemented
32#19149661—21no—
33#19149661—21no—
34#19149658C2211yescharacteristic
35#19149658C2211yescharacteristic, complemented
36#19149650—21no—
37C4×D692C2211yescharacteristic, complemented
38C4⋊C4×D173—21no—
39#19149610C2211yescharacteristic, complemented
40C4⋊#13063436C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149602C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149600C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149599C2211yescharacteristic, complemented
47C22⋊#13063436C2211yescharacteristic
48C22⋊#13063436C2211yescharacteristic, complemented
49#19149590C2211yescharacteristic, complemented
50C2×#41×C173C2211yescharacteristic
51C22×D4×C173C2211yescharacteristic
52#124×C173C2211yescharacteristic
53#123×C173—21no—
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173—21no—
56#121×C173C2211yescharacteristic
57#120×C173—21no—
58C4×D4×C173C2211yescharacteristic
59C4×D4×C173C2211yescharacteristic
60C2×C22⋊C4×C173—21no—
61#14583826—22no—
62#14583826—22no—
63#14583826—41no—
64#14583826—41no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73#14583816—22no—
74#14583816—41no—
75#14583814—22no—
76#14583814—22no—
77#14583814—22no—
78#14583814—41no—
79#14583813—22no—
80#14583811—22no—
81#14583811D414yescomplemented
82#14583810—22no—
83C2×#13063436—22no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173—21no—
89C4⋊Dic173—21no—
90C4⋊Dic173C2311yescharacteristic
91#14583805—21no—
92#14583805—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95#14583803—21no—
96C24×C173—21no—
97#41×C173—22no—
98#41×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—22no—
101C2×D4×C173—21no—
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173—41no—
105C2×D4×C173D411yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C2×D4×C173C2311yescharacteristic
108C22×C4×C173—21no—
109C22×C4×C173—21no—
110C22×C4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112C4⋊C4×C173—21no—
113C4⋊C4×C173D411yescharacteristic
114C4⋊C4×C173C2311yescharacteristic
115C22⋊C4×C173—21no—
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128D692—21no—
129D692C2×D412yes—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C4×D173—42no—
136C4×D173—42no—
137C4×D173—42no—
138#13063436—21no—
139#13063436—21no—
140#13063436—21no—
141#13063436C2×D412yes—
142C23×C173—21no—
143C23×C173—21no—
144C23×C173—42no—
145C23×C173—42no—
146C23×C173—81no—
147C23×C173C2×D411yescharacteristic
148C23×C173C2×D411yescharacteristic
149Q8×C173—21no—
150Q8×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—41no—
155D4×C173—42no—
156D4×C173—42no—
157D4×C173—81no—
158D4×C173C2×D412yes—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—42no—
166C2×C4×C173—81no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2×D411yescharacteristic
170C2×C4×C173C2×D411yescharacteristic
171C2×C4×C173C2411yescharacteristic
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—22no—
175C22×C173—21no—
176C22×C173—21no—
177C22×C173—21no—
178C22×C173—21no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—84no—
185C22×C173C22×D411yescharacteristic
186C2×D173—22no—
187C2×D173—41no—
188C2×D173—41no—
189C4×C173—21no—
190C4×C173—21no—
191C4×C173—21no—
192C4×C173—21no—
193C4×C173—41no—
194C4×C173—41no—
195C4×C173C22×D411yescharacteristic
196C4×C173H(5,𝔽2)11yescharacteristic
197Dic173—41no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—41no—
201C2×C173—41no—
202C2×C173—42no—
203C2×C173—81no—
204C2×C173D4211yescharacteristic
205C2×C173#57511yescharacteristic
206C2×C173#57711yescharacteristic
207D173—42no—
208C173#326711yescharacteristic, complemented, Sylow
209#3267—1731nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1731no—
211C2×#135—1731no—
212#547—1731no—
213#546D17311yescharacteristic, complemented
214#497—1731no—
215#470—1731no—
216#465—1731no—
217#463—1731no—
218#461—1731no—
219#452—1731no—
220#452—1731no—
221#450—1731no—
222#449—1731no—
223#442—1731no—
224#435—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C2×#41C2×D17311yescharacteristic
228C22×D4C2×D17311yescharacteristic
229AGL(1,ℤ/8)—3464no—
230#135—3464no—
231C2×Q16—1731no—
232C2×Q16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×SD16—1731no—
237C2×SD16—1731no—
238C2×D8—1731no—
239C2×D8—1731no—
240C2×M16—1731no—
241C22×C8—1731no—
242C4⋊Q8—1731no—
243#124C2×D17311yescharacteristic
244#123—21no—
245#123—3461no—
246#123—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8—1731no—
249C22⋊Q8C2×D17311yescharacteristic
250#121—21no—
251#121—3461no—
252#121C2×D17311yescharacteristic
253#120—21no—
254C4×D4—1731no—
255C4×D4C2×D17311yescharacteristic
256C4×D4C2×D17311yescharacteristic
257C2×C4⋊C4—3461no—
258C2×C22⋊C4—21no—
259#106—1731no—
260C4⋊C8—1731no—
261#103—1731no—
262#103—1731no—
263#103—1731no—
264#102—1731no—
265#102—1731no—
266#102—1731no—
267C22⋊C8—1731no—
268C22⋊C8—1731no—
269#97—1731no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—3462no—
274#41—3462no—
275#41—6921no—
276#41—6921no—
277C2×Q8—1731no—
278C2×Q8—1731no—
279C2×Q8C22×D17311yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—1731no—
288C2×D4C22⋊D17311yescharacteristic
289C2×D4C22×D17311yescharacteristic
290C2×D4C22×D17311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4C22×D17311yescharacteristic
297C22×C4C22×D17311yescharacteristic
298Q16—3462no—
299Q16—6921no—
300SD16—3462no—
301SD16—3462no—
302SD16—3462no—
303SD16—3462no—
304SD16—6921no—
305D8—3462no—
306D8—1734no—
307M16—3462no—
308C2×C8—3462no—
309C2×C8—1731no—
310C2×C8—1731no—
311C2×C8—1731no—
312C2×C8—1731no—
313C4⋊C4—21no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—3461no—
317C4⋊C4—3461no—
318C4⋊C4—1731no—
319C4⋊C4C22⋊D17311yescharacteristic
320C4⋊C4C22×D17311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4C22⋊D17311yescharacteristic
330C22⋊C4C22⋊D17311yescharacteristic
331C42C22×D17311yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—3461no—
338C23C2×C22⋊D17311yescharacteristic
339C23C2×C22⋊D17311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3461no—
343Q8—3461no—
344Q8—3461no—
345Q8—3461no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—3461no—
354D4—1732no—
355D4—6922no—
356D4—6922no—
357D4D4×D17312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—6922no—
370C2×C4—6922no—
371C2×C4—6922no—
372C2×C4C2×C22⋊D17311yescharacteristic
373C2×C4C2×C22⋊D17311yescharacteristic
374C2×C4C2×C22⋊D17311yescharacteristic
375C2×C4C2×C22⋊D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—3461no—
378C8—3461no—
379C8—3461no—
380C8—1732no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3462no—
395C22—6921no—
396C22—6921no—
397C22C22×C22⋊D17311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—6921no—
405C4—6921no—
406C4—6921no—
407C4C2×D4×D17311yescharacteristic
408C4#1914972411yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—6922no—
414C2D4×C22⋊D17311yescharacteristic
415C2#2601228911yescharacteristic
416C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#546×C173C2
socleC22×C173C22×D4
hypercenter#546D173
hypocenterC173#3267
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C2×#19149699C2semidirect
3. #26012146C2semidirect
4. #26011886C2semidirect
5. #26011747C2semidirect
6. #26011718C2semidirect
7. #26011703C2semidirect
8. #26011696C2semidirect
9. #26011653C2semidirect
10. #26011651C2semidirect
11. #26011642C2semidirect
12. #26011634C2semidirect
13. #26011601C2semidirect
14. #26011563C2semidirect
15. #546×C173C2semidirect
17. C2×#14583826C22semidirect
20. C2×#14583816C22semidirect
21. C2×#14583816C22semidirect
26. C2×#14583813C22semidirect
30. C22×#13063436C22semidirect
31. #19149679C22semidirect
35. #19149658C22semidirect
37. C4×D692C22semidirect
39. #19149610C22semidirect
40. C4⋊#13063436C22semidirect
41. #19149603C22semidirect
42. #19149602C22semidirect
43. #19149602C22semidirect
46. #19149599C22semidirect
48. C22⋊#13063436C22semidirect
49. #19149590C22semidirect
81. #14583811D4semidirect
208. C173#3267semidirect
213. #546D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C2×#191496992
#260121462
#260118862
#260117472
#260117182
#260117032
#260116962
#260116532
#260116512
#260116422
#260116342
#260116012
#260115632
#546×C1732
#3267173