Finite Group Databasebeta

← #36365086

Subgroups — #36365086

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×#19149700C211yesmaximal, characteristic, complemented
3#26012143C211yesmaximal, characteristic, complemented
4#26012135C211yesmaximal, characteristic, complemented
5#26011882C211yesmaximal, characteristic, complemented
6#26011747C211yesmaximal, characteristic, complemented
7#26011719C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011694C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011654C211yesmaximal, characteristic, complemented
12#26011635C211yesmaximal, characteristic, complemented
13#26011600C211yesmaximal, characteristic, complemented
14#26011565C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic, complemented
17C2×#14583826C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19#19149705—24no—
20#19149700—24no—
21C2×Dic692C2211yescharacteristic, complemented
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C2×#14583810C2211yescharacteristic
29C2×C8×D173C2211yescharacteristic, complemented
30#19149677C2211yescharacteristic, complemented
31#19149666C2211yescharacteristic
32#19149664—21no—
33#19149661—21no—
34#19149660—21no—
35#19149659C2211yescharacteristic, complemented
36#19149658C2211yescharacteristic
37#19149652—21no—
38#19149651—21no—
39#19149650C2211yescharacteristic
40#19149648—21no—
41D4×Dic173C2211yescharacteristic
42#19149641C2211yescharacteristic, complemented
43C4×C22⋊D173C2211yescharacteristic
44C4⋊C4×D173—21no—
45C2×C22⋊Dic173—21no—
46#19149610C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149589C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C2×SD16×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—41no—
63#14583826—22no—
64#14583826—41no—
65#14583825—22no—
66#14583825—41no—
67C2×Dic346C2311yescharacteristic
68C2×Dic346C2311yescharacteristic
69C2×D692—21no—
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C22×Dic173—21no—
76C22×Dic173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173C2311yescharacteristic
81Dic692—41no—
82#14583816—22no—
83#14583815—22no—
84#14583814—22no—
85#14583814—22no—
86#14583811—22no—
87#14583811D414yescomplemented
88#14583810—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C8×D173—22no—
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94#14583805—21no—
95#14583805—21no—
96#14583805—21no—
97#14583805D411yescharacteristic
98C22⋊Dic173—21no—
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107C4×Dic173C2311yescharacteristic
108C24×C173—21no—
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173—22no—
111C2×D4×C173—21no—
112C2×D4×C173—41no—
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173C2311yescharacteristic
116SD16×C173—22no—
117SD16×C173—41no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173—21no—
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129D692—21no—
130D692—42no—
131D692C2×D412yes—
132C22⋊D173—41no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—81no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173—81no—
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C4×D173—21no—
145C4×D173—21no—
146C4×D173—42no—
147C4×D173—42no—
148C4×D173—42no—
149#13063436—21no—
150#13063436C2×D412yes—
151C23×C173—21no—
152C23×C173—21no—
153C23×C173—42no—
154C23×C173—42no—
155C23×C173—81no—
156C23×C173C2×D411yescharacteristic
157Q8×C173—21no—
158Q8×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—42no—
163D4×C173C2×D412yes—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173—21no—
172C22×C173—21no—
173C22×C173—22no—
174C22×C173—21no—
175C22×C173—21no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—84no—
182C22×C173C22×D411yescharacteristic
183C2×D173—21no—
184C2×D173—22no—
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—41no—
188C2×D173—41no—
189C4×C173—21no—
190C4×C173—41no—
191C4×C173—41no—
192C4×C173C22×D411yescharacteristic
193C4×C173H(5,𝔽2)11yescharacteristic
194Dic173—21no—
195Dic173—21no—
196Dic173—21no—
197Dic173—41no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—41no—
201C2×C173—42no—
202C2×C173—81no—
203C2×C173D4211yescharacteristic
204C2×C173#57511yescharacteristic
205C2×C173#57711yescharacteristic
206D173—41no—
207D173—42no—
208C173#326711yescharacteristic, complemented, Sylow
209#3267—1731nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1731no—
211C2×#135—1731no—
212#547—1731no—
213#546—1731no—
214#497—1731no—
215#470—1731no—
216#465—1731no—
217#463—1731no—
218#461—1731no—
219#452—1731no—
220#452—1731no—
221#450D17311yescharacteristic, complemented
222#449—1731no—
223#442—1731no—
224#435—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C2×#41—1731no—
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×SD16C2×D17311yescharacteristic
237C2×SD16C2×D17311yescharacteristic
238C2×D8—1731no—
239C2×D8—1731no—
240C2×M16—1731no—
241C22×C8—1731no—
242C4⋊Q8—1731no—
243#124—1731no—
244#123—3461no—
245#123—3461no—
246#123—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8—1731no—
249C22⋊Q8C2×D17311yescharacteristic
250#121—3461no—
251#121—3461no—
252#121—1731no—
253#120—3461no—
254C4×D4—1731no—
255C4×D4—1731no—
256C4×D4—1731no—
257C2×C4⋊C4—3461no—
258C2×C22⋊C4—3461no—
259#106—1731no—
260C4⋊C8—1731no—
261#103—1731no—
262#103—1731no—
263#103—1731no—
264#102—1731no—
265#102C2×D17311yescharacteristic
266#102C2×D17311yescharacteristic
267C22⋊C8—1731no—
268C22⋊C8C2×D17311yescharacteristic
269#97—1731no—
270C24—21no—
271#41—3462no—
272#41—3462no—
273#41—3462no—
274#41—6921no—
275#41—6921no—
276#41—6921no—
277C2×Q8—1731no—
278C2×Q8—1731no—
279C2×Q8C22×D17311yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—1731no—
288C2×D4—1731no—
289C2×D4C22×D17311yescharacteristic
290C2×D4C22×D17311yescharacteristic
291C22×C4—3461no—
292C22×C4—3461no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4C22×D17311yescharacteristic
298Q16—3462no—
299Q16—6921no—
300SD16—22no—
301SD16—41no—
302SD16—3462no—
303SD16—3462no—
304SD16—3462no—
305D8—3462no—
306D8—1734no—
307M16—3462no—
308C2×C8—3462no—
309C2×C8—1731no—
310C2×C8—1731no—
311C2×C8C22×D17311yescharacteristic
312C2×C8C22×D17311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—3461no—
317C4⋊C4—3461no—
318C4⋊C4—1731no—
319C4⋊C4—1731no—
320C4⋊C4C22×D17311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—1731no—
330C22⋊C4—1731no—
331C42—1731no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—3461no—
338C23—1731no—
339C23D4×D17311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3461no—
343Q8—3461no—
344Q8—3461no—
345Q8—3461no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—3461no—
351D4—13841no—
352D4—1732no—
353D4—6922no—
354D4—6921no—
355D4—6922no—
356D4—6922no—
357D4D4×D17312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—13841no—
368C2×C4—1731no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—6922no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4—6922no—
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—21no—
379C8—3461no—
380C8—1732no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—84no—
391C22—3461no—
392C22—3461no—
393C22—3461no—
394C22—3462no—
395C22—6921no—
396C22—6921no—
397C22C2×D4×D17311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—3461no—
402C4—3461no—
403C4—3461no—
404C4—6921no—
405C4—6921no—
406C4—6921no—
407C4C2×D4×D17311yescharacteristic
408C4#1914972411yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—6921no—
413C2—6922no—
414C2#2601212311yescharacteristic
415C2#2601228711yescharacteristic
416C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
hypercenter#450D173
hypocenterC173#3267
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C2×#19149700C2semidirect
3. #26012143C2semidirect
4. #26012135C2semidirect
5. #26011882C2semidirect
6. #26011747C2semidirect
7. #26011719C2semidirect
8. #26011705C2semidirect
9. #26011694C2semidirect
10. #26011655C2semidirect
11. #26011654C2semidirect
12. #26011635C2semidirect
13. #26011600C2semidirect
14. #26011565C2semidirect
15. #450×C173C2semidirect
16. C2×#14583827C22semidirect
21. C2×Dic692C22semidirect
22. C2×#14583816C22semidirect
23. C2×#14583815C22semidirect
29. C2×C8×D173C22semidirect
30. #19149677C22semidirect
35. #19149659C22semidirect
42. #19149641C22semidirect
46. #19149610C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149603C22semidirect
50. #19149602C22semidirect
51. #19149601C22semidirect
52. #19149591C22semidirect
53. #19149589C22semidirect
87. #14583811D4semidirect
208. C173#3267semidirect
221. #450D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C2×#191497002
#260121432
#260121352
#260118822
#260117472
#260117192
#260117052
#260116942
#260116552
#260116542
#260116352
#260116002
#260115652
#450×C1732
#3267173