Finite Group Databasebeta

← #36365110

Subgroups — #36365110

Order 22144.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149707C211yesmaximal, characteristic, complemented
2C2×#19149700C211yesmaximal, characteristic, complemented
3#26012159C211yesmaximal, characteristic, complemented
4#26012132C211yesmaximal, characteristic, complemented
5#26011877C211yesmaximal, characteristic, complemented
6#26011728C211yesmaximal, characteristic, complemented
7#26011716C211yesmaximal, characteristic, complemented
8#26011685C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011639C211yesmaximal, characteristic, complemented
11#26011637C211yesmaximal, characteristic, complemented
12#26011624C211yesmaximal, characteristic, complemented
13#26011590C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583825C2211yescharacteristic, complemented
19C22×D692C2211yescharacteristic
20#19149707—24no—
21#19149700—24no—
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic
24C2×#14583815C2211yescharacteristic
25C2×#14583813C2211yescharacteristic, complemented
26C2×D1384C2211yescharacteristic
27C2×D1384C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583811C2211yescharacteristic
30C2×C8×D173C2211yescharacteristic, complemented
31#19149675—21no—
32#19149675C2211yescharacteristic, complemented
33#19149668C2211yescharacteristic
34#19149664—21no—
35#19149654—21no—
36#19149654C2211yescharacteristic
37#19149654C2211yescharacteristic, complemented
38#19149652—21no—
39#19149651—21no—
40#19149649—21no—
41#19149644C2211yescharacteristic, complemented
42#19149641C2211yescharacteristic
43C4×C22⋊D173C2211yescharacteristic
44#19149639—21no—
45C22⋊C4×D173—21no—
46#19149608C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149599C2211yescharacteristic, complemented
51C22⋊#13063436C2211yescharacteristic
52#19149591C2211yescharacteristic, complemented
53#19149589C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×M16×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64#14583826—22no—
65#14583826—41no—
66#14583825—22no—
67#14583825—41no—
68C2×Dic346C2311yescharacteristic
69C2×D692—22no—
70C2×D692—21no—
71C2×D692—21no—
72C2×D692—21no—
73C2×D692—21no—
74C2×D692—41no—
75C2×D692C2311yescharacteristic
76C2×D692C2311yescharacteristic
77C2×D692C2311yescharacteristic
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80C2×C22⋊D173—21no—
81C2×C22⋊D173—21no—
82C2×C22⋊D173D411yescharacteristic
83C22×Dic173—21no—
84C2×C4×D173—21no—
85C2×C4×D173—21no—
86C2×C4×D173—21no—
87C2×C4×D173C2311yescharacteristic
88#14583816—22no—
89#14583815—22no—
90#14583815D414yescomplemented
91#14583813—41no—
92D1384—22no—
93D1384—22no—
94#14583811—22no—
95#14583811—41no—
96C2×#13063436C2311yescharacteristic
97C2×#13063436C2311yescharacteristic
98C8×D173—22no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173D411yescharacteristic
102#14583803—21no—
103#14583803—21no—
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—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—81no—
129C22×D173C2×D411yescharacteristic
130Dic346—21no—
131Dic346C2×D412yes—
132D692—21no—
133D692—21no—
134D692—21no—
135D692—21no—
136D692—21no—
137D692—42no—
138D692—42no—
139D692—81no—
140D692—81no—
141D692C2×D412yes—
142C22⋊D173—41no—
143C22⋊D173—42no—
144C22⋊D173—42no—
145C22⋊D173—42no—
146C22⋊D173—81no—
147C2×Dic173—21no—
148C2×Dic173—42no—
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C2×Dic173C2×D411yescharacteristic
152C4×D173—21no—
153C4×D173—21no—
154C4×D173—42no—
155C4×D173—42no—
156C4×D173—81no—
157#13063436—21no—
158#13063436—21no—
159C23×C173—21no—
160C23×C173C2×D411yescharacteristic
161Q8×C173—21no—
162Q8×C173—21no—
163D4×C173—21no—
164D4×C173—21no—
165D4×C173—81no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173C2×D412yes—
175C22×C173—21no—
176C22×C173—21no—
177C22×C173—21no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173C22×D411yescharacteristic
182C2×D173—21no—
183C2×D173—22no—
184C2×D173—21no—
185C2×D173—21no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—84no—
195C4×C173—21no—
196C4×C173—41no—
197C4×C173—41no—
198C4×C173—41no—
199C4×C173C22×D411yescharacteristic
200C4×C173H(5,𝔽2)11yescharacteristic
201Dic173—21no—
202Dic173—22no—
203Dic173—21no—
204Dic173—21no—
205C2×C173—41no—
206C2×C173—81no—
207C2×C173D4211yescharacteristic
208C2×C173#57511yescharacteristic
209C2×C173#57611yescharacteristic
210D173—41no—
211D173—42no—
212D173—81no—
213D173—81no—
214C173#326811yescharacteristic, complemented, Sylow
215#3268—1731nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1731no—
217C2×#135—1731no—
218#550—1731no—
219#546—1731no—
220#496—1731no—
221#468D17311yescharacteristic, complemented
222#466—1731no—
223#464—1731no—
224#459—1731no—
225#452—1731no—
226#449—1731no—
227#449—1731no—
228#447—1731no—
229#440—1731no—
230#435—1731no—
231C2×#41—1731no—
232C2×#41—1731no—
233C2×#41—1731no—
234C22×D4—1731no—
235AGL(1,ℤ/8)—3464no—
236#135—3464no—
237C2×Q16—1731no—
238C2×SD16—1731no—
239C2×SD16—1731no—
240C2×SD16—1731no—
241C2×SD16C2×D17311yescharacteristic, complemented
242C2×D8—1731no—
243C2×D8—1731no—
244C2×D8—1731no—
245C2×D8—1731no—
246C2×M16C2×D17311yescharacteristic
247C22×C8—1731no—
248#127—3461no—
249#127—1731no—
250#124—1731no—
251#123—3461no—
252C22⋊Q8C2×D17311yescharacteristic
253#121—3461no—
254#121—3461no—
255#121—3461no—
256#121—1731no—
257#121—1731no—
258#121C2×D17311yescharacteristic
259#120—3461no—
260C4×Q8—1731no—
261C4×D4—3461no—
262C4×D4—1731no—
263C4×D4—1731no—
264C2×C22⋊C4—3461no—
265#107C2×D17311yescharacteristic, complemented
266C4⋊C8—1731no—
267#103—1731no—
268#103C2×D17311yescharacteristic, complemented
269#102—1731no—
270#102—1731no—
271#102—1731no—
272#102C2×D17311yescharacteristic, complemented
273C22⋊C8—1731no—
274C22⋊C8—1731no—
275#97—1731no—
276C24—3461no—
277#41—3462no—
278#41—3462no—
279#41—3462no—
280#41—6921no—
281#41—6921no—
282#41—6921no—
283C2×Q8—1731no—
284C2×Q8C22×D17311yescharacteristic
285C2×D4—21no—
286C2×D4—3462no—
287C2×D4—3461no—
288C2×D4—3461no—
289C2×D4—3461no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—3461no—
294C2×D4—3461no—
295C2×D4—1731no—
296C2×D4—1731no—
297C2×D4—1731no—
298C2×D4—1731no—
299C2×D4—6921no—
300C2×D4C22×D17311yescharacteristic
301C22×C4—3461no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4—3461no—
305C22×C4—1731no—
306C22×C4C22×D17311yescharacteristic
307Q16—3462no—
308SD16—22no—
309SD16—3462no—
310SD16—1734no—
311SD16—6921no—
312D8—3462no—
313D8—3462no—
314D8—3462no—
315D8—6921no—
316M16—22no—
317C2×C8—3462no—
318C2×C8—1731no—
319C2×C8—1731no—
320C2×C8C22×D17311yescharacteristic
321C2×C8C22×D17311yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—3461no—
324C4⋊C4—1731no—
325C4⋊C4C22×D17311yescharacteristic
326C4⋊C4C22×D17311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—1731no—
335C22⋊C4—1731no—
336C42—3461no—
337C42—1731no—
338C23—21no—
339C23—3461no—
340C23—3461no—
341C23—3461no—
342C23—13841no—
343C23—1731no—
344C23—6922no—
345C23—6922no—
346C23D4×D17311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—3461no—
350Q8—1732no—
351D4—21no—
352D4—21no—
353D4—81no—
354D4—3461no—
355D4—3461no—
356D4—3461no—
357D4—3461no—
358D4—3461no—
359D4—13841no—
360D4—13841no—
361D4—13841no—
362D4—1732no—
363D4—6922no—
364D4—6922no—
365D4—6921no—
366D4—6922no—
367D4—6922no—
368D4—6922no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—13841no—
378C2×C4—1731no—
379C2×C4—1731no—
380C2×C4—1731no—
381C2×C4—6922no—
382C2×C4—6922no—
383C2×C4—6922no—
384C2×C4D4×D17311yescharacteristic
385C2×C4C23×D17311yescharacteristic
386C8—21no—
387C8—3461no—
388C8—3461no—
389C8D4×D17312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—3461no—
397C22—3461no—
398C22—3461no—
399C22—3462no—
400C22—13844no—
401C22—6921no—
402C22—6921no—
403C22—6921no—
404C22—6921no—
405C22—6921no—
406C22—6921no—
407C22—6921no—
408C22—6921no—
409C22C2×D4×D17311yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—3461no—
415C4—3461no—
416C4—3461no—
417C4—3462no—
418C4C2×D4×D17311yescharacteristic
419C4#1914972511yescharacteristic
420C2—41no—
421C2—81no—
422C2—13841no—
423C2—13841no—
424C2—6921no—
425C2—6922no—
426C2#2601212511yescharacteristic
427C2#2601229011yescharacteristic
428C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3268
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×#19149707C2semidirect
2. C2×#19149700C2semidirect
3. #26012159C2semidirect
4. #26012132C2semidirect
5. #26011877C2semidirect
6. #26011728C2semidirect
7. #26011716C2semidirect
8. #26011685C2semidirect
9. #26011651C2semidirect
10. #26011639C2semidirect
11. #26011637C2semidirect
12. #26011624C2semidirect
13. #26011590C2semidirect
14. #26011560C2semidirect
15. #468×C173C2semidirect
18. C2×#14583825C22semidirect
22. C2×#14583816C22semidirect
25. C2×#14583813C22semidirect
30. C2×C8×D173C22semidirect
32. #19149675C22semidirect
37. #19149654C22semidirect
41. #19149644C22semidirect
46. #19149608C22semidirect
47. #19149602C22semidirect
50. #19149599C22semidirect
52. #19149591C22semidirect
53. #19149589C22semidirect
54. C2×SD16×C173C22semidirect
58. #107×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
90. #14583815D4semidirect
214. C173#3268semidirect
221. #468D173semidirect
241. C2×SD16C2×D173semidirect
265. #107C2×D173semidirect
268. #103C2×D173semidirect
272. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
C2×#191497002
#260121592
#260121322
#260118772
#260117282
#260117162
#260116852
#260116512
#260116392
#260116372
#260116242
#260115902
#260115602
#468×C1732
#3268173