Finite Group Databasebeta

← #36365181

Subgroups — #36365181

Order 22144.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149705C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3#26012159C211yesmaximal, characteristic, complemented
4D4×C22⋊D173C211yesmaximal, characteristic, complemented
5#26011866C211yesmaximal, characteristic, complemented
6#26011743C211yesmaximal, characteristic, complemented
7#26011742C211yesmaximal, characteristic, complemented
8#26011690C211yesmaximal, characteristic, complemented
9#26011684C211yesmaximal, characteristic, complemented
10#26011638C211yesmaximal, characteristic, complemented
11#26011633C211yesmaximal, characteristic, complemented
12#26011626C211yesmaximal, characteristic, complemented
13#26011582C211yesmaximal, characteristic, complemented
14#26011566C211yesmaximal, characteristic, complemented
15#447×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C2×#14583825C2211yescharacteristic, complemented
18C22×C22⋊D173—21no—
19C2×D4×D173C2211yescharacteristic
20#19149705—24no—
21#19149703—24no—
22C2×#14583815C2211yescharacteristic, complemented
23C2×#14583814C2211yescharacteristic
24C2×#14583814C2211yescharacteristic, complemented
25C2×D1384C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic, complemented
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583810C2211yescharacteristic, complemented
30C2×#14583809C2211yescharacteristic, complemented
31#19149675—21no—
32#19149675C2211yescharacteristic
33#19149675C2211yescharacteristic, complemented
34#19149654C2211yescharacteristic, complemented
35#19149653C2211yescharacteristic
36#19149652—21no—
37#19149651—21no—
38#19149651—21no—
39#19149650C2211yescharacteristic
40#19149648—21no—
41#19149647—21no—
42#19149644C2211yescharacteristic, complemented
43D4×Dic173C2211yescharacteristic
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149610C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149600C2211yescharacteristic, complemented
51#19149599C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53C8×Dic173C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56C2×D8×C173C2211yescharacteristic, complemented
57#121×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic, complemented
60C22⋊C8×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583826—22no—
63#14583826—41no—
64#14583825—22no—
65#14583825—41no—
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C2×C22⋊D173—44no—
80C2×C22⋊D173D411yescharacteristic
81D4×D173—22no—
82D4×D173—41no—
83C22×Dic173—21no—
84C22×Dic173—21no—
85C2×C4×D173—21no—
86C2×C4×D173C2311yescharacteristic
87#14583815—22no—
88#14583814—22no—
89#14583814D414yescomplemented
90D1384—41no—
91#14583811—22no—
92#14583811—22no—
93#14583811—22no—
94#14583810—22no—
95#14583809—22no—
96C2×#13063436C2311yescharacteristic
97C2×#13063436C2311yescharacteristic
98C4⋊Dic173C2311yescharacteristic
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C4×Dic173C2311yescharacteristic
109C24×C173—21no—
110C2×D4×C173—22no—
111C2×D4×C173—21no—
112C2×D4×C173—21no—
113C2×D4×C173—41no—
114C2×D4×C173C2311yescharacteristic
115C2×D4×C173C2311yescharacteristic
116C2×D4×C173C2311yescharacteristic
117C22×C4×C173C2311yescharacteristic
118D8×C173—22no—
119D8×C173—41no—
120C2×C8×C173C2311yescharacteristic
121C2×C8×C173C2311yescharacteristic
122C4⋊C4×C173C2311yescharacteristic
123C22⋊C4×C173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—81no—
129C22×D173C2×D411yescharacteristic
130Dic346—21no—
131Dic346C2×D412yescomplemented
132D692—21no—
133D692—21no—
134D692—21no—
135D692—21no—
136D692—81no—
137C22⋊D173—22no—
138C22⋊D173—42no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—42no—
142C22⋊D173—42no—
143C22⋊D173—42no—
144C22⋊D173—42no—
145C22⋊D173—81no—
146C22⋊D173—81no—
147C2×Dic173—21no—
148C2×Dic173—21no—
149C2×Dic173—42no—
150C2×Dic173—42no—
151C2×Dic173C2×D411yescharacteristic
152C2×Dic173C2×D411yescharacteristic
153C2×Dic173C2×D411yescharacteristic
154C4×D173—21no—
155C4×D173—21no—
156C4×D173—42no—
157#13063436—21no—
158#13063436C2×D412yescomplemented
159C23×C173—21no—
160C23×C173—21no—
161C23×C173—21no—
162C23×C173—42no—
163C23×C173—42no—
164C23×C173—81no—
165C23×C173C2×D411yescharacteristic
166D4×C173—21no—
167D4×C173—21no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173—21no—
171D4×C173—42no—
172D4×C173—81no—
173D4×C173C2×D412yes—
174C2×C4×C173—21no—
175C2×C4×C173—21no—
176C2×C4×C173—21no—
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2411yescharacteristic
179C8×C173—21no—
180C8×C173—21no—
181C22×C173—21no—
182C22×C173—22no—
183C22×C173—21no—
184C22×C173—21no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173—41no—
191C22×C173—41no—
192C22×C173—41no—
193C22×C173—84no—
194C22×C173C22×D411yescharacteristic
195C2×D173—21no—
196C2×D173—21no—
197C2×D173—21no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—41no—
204C2×D173—84no—
205C4×C173—21no—
206C4×C173—41no—
207C4×C173C22×D411yescharacteristic
208C4×C173H(5,𝔽2)11yescharacteristic
209Dic173—21no—
210Dic173—22no—
211Dic173—21no—
212Dic173—41no—
213Dic173AGL(1,ℤ/8)12yescomplemented
214C2×C173—42no—
215C2×C173—41no—
216C2×C173—81no—
217C2×C173—81no—
218C2×C173D4211yescharacteristic
219C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C173#57611yescharacteristic
221D173—41no—
222D173—81no—
223D173—81no—
224C173#327411yescharacteristic, complemented, Sylow
225#3274—1731nomaximal, Sylow
226C2×AGL(1,ℤ/8)—1731no—
227C2×AGL(1,ℤ/8)—1731no—
228#550—1731no—
229D42—1731no—
230#493—1731no—
231#469—1731no—
232#469—1731no—
233#460—1731no—
234#459—1731no—
235#449—1731no—
236#449—1731no—
237#447—1731no—
238#447D17311yescharacteristic, complemented
239C4×SD16—1731no—
240#436—1731no—
241C2×#41—1731no—
242C2×#41—1731no—
243C22×D4—3461no—
244C22×D4—1731no—
245C22×D4C2×D17311yescharacteristic
246AGL(1,ℤ/8)—3464no—
247AGL(1,ℤ/8)—3464no—
248C2×SD16—1731no—
249C2×SD16—1731no—
250C2×SD16—1731no—
251C2×D8—1731no—
252C2×D8—1731no—
253C2×D8—1731no—
254C2×D8—1731no—
255C2×D8C2×D17311yescharacteristic
256C2×D8C2×D17311yescharacteristic, complemented
257C2×M16—1731no—
258C2×M16—1731no—
259#127—3461no—
260#127—1731no—
261#127—1731no—
262#121—3461no—
263#121—3461no—
264#121—3461no—
265#121—1731no—
266#121—1731no—
267#121—1731no—
268#121C2×D17311yescharacteristic, complemented
269#120—3461no—
270#120—3461no—
271C4×Q8—1731no—
272C4×D4—3461no—
273C4×D4—1731no—
274C4×D4—1731no—
275#106—1731no—
276C4⋊C8—1731no—
277#103—1731no—
278#102—1731no—
279#102—1731no—
280#102—1731no—
281#102C2×D17311yescharacteristic
282#102C2×D17311yescharacteristic, complemented
283C22⋊C8—1731no—
284C22⋊C8C2×D17311yescharacteristic, complemented
285C4×C8—1731no—
286C24—21no—
287C24—3461no—
288#41—3462no—
289#41—3462no—
290#41—6921no—
291#41—6921no—
292C2×Q8—1731no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—3461no—
298C2×D4—3462no—
299C2×D4—3461no—
300C2×D4—3461no—
301C2×D4—3461no—
302C2×D4—3461no—
303C2×D4—3461no—
304C2×D4—3461no—
305C2×D4—3461no—
306C2×D4—3461no—
307C2×D4—3461no—
308C2×D4—1731no—
309C2×D4—1731no—
310C2×D4—1731no—
311C2×D4—6921no—
312C2×D4—6924no—
313C2×D4C22×D17311yescharacteristic
314C2×D4C22×D17311yescharacteristic
315C2×D4C22×D17311yescharacteristic
316C22×C4—3461no—
317C22×C4—3461no—
318C22×C4—3461no—
319C22×C4—1731no—
320C22×C4C22×D17311yescharacteristic
321SD16—3462no—
322SD16—3462no—
323SD16—1734no—
324D8—22no—
325D8—41no—
326D8—3462no—
327D8—3462no—
328D8—3462no—
329D8—6921no—
330M16—3462no—
331M16—3462no—
332C2×C8—1731no—
333C2×C8—1731no—
334C2×C8C22×D17311yescharacteristic
335C2×C8C22×D17311yescharacteristic
336C4⋊C4—3461no—
337C4⋊C4—1731no—
338C4⋊C4—1731no—
339C4⋊C4C22×D17311yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—3461no—
342C22⋊C4—3461no—
343C22⋊C4—3461no—
344C22⋊C4—3461no—
345C22⋊C4—1731no—
346C22⋊C4—1731no—
347C42—3461no—
348C42—1731no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—3461no—
356C23—3461no—
357C23—13841no—
358C23—1731no—
359C23—6922no—
360C23—6922no—
361C23D4×D17311yescharacteristic
362Q8—3461no—
363Q8—1732no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—81no—
371D4—3462no—
372D4—3461no—
373D4—3461no—
374D4—3461no—
375D4—3461no—
376D4—13841no—
377D4—13841no—
378D4—13841no—
379D4—6922no—
380D4—6922no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4D4×D17312yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—1731no—
395C2×C4—1731no—
396C2×C4—1731no—
397C2×C4—6922no—
398C2×C4—6922no—
399C2×C4—6922no—
400C2×C4D4×D17311yescharacteristic
401C2×C4C23×D17311yescharacteristic
402C8—21no—
403C8—21no—
404C8—3461no—
405C8—1732no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—3461no—
420C22—3461no—
421C22—3461no—
422C22—13844no—
423C22—6921no—
424C22—6921no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22C2×D4×D17311yescharacteristic
430C4—21no—
431C4—41no—
432C4—3461no—
433C4—3461no—
434C4—3462no—
435C4—1732no—
436C4—6921no—
437C4C2×D4×D17311yescharacteristic
438C4#1914972411yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—13841no—
444C2—13841no—
445C2—6921no—
446C2#2601212311yescharacteristic
447C2AGL(1,ℤ/8)×D17311yescharacteristic
448C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C2×#19149703C2semidirect
3. #26012159C2semidirect
4. D4×C22⋊D173C2semidirect
5. #26011866C2semidirect
6. #26011743C2semidirect
7. #26011742C2semidirect
8. #26011690C2semidirect
9. #26011684C2semidirect
10. #26011638C2semidirect
11. #26011633C2semidirect
12. #26011626C2semidirect
13. #26011582C2semidirect
14. #26011566C2semidirect
15. #447×C173C2semidirect
16. C2×#14583826C22semidirect
17. C2×#14583825C22semidirect
22. C2×#14583815C22semidirect
24. C2×#14583814C22semidirect
25. C2×D1384C22semidirect
27. C2×#14583811C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583810C22semidirect
30. C2×#14583809C22semidirect
33. #19149675C22semidirect
34. #19149654C22semidirect
42. #19149644C22semidirect
46. #19149610C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149601C22semidirect
50. #19149600C22semidirect
51. #19149599C22semidirect
52. #19149591C22semidirect
53. C8×Dic173C22semidirect
56. C2×D8×C173C22semidirect
57. #121×C173C22semidirect
59. #102×C173C22semidirect
60. C22⋊C8×C173C22semidirect
89. #14583814D4semidirect
131. Dic346C2×D4semidirect
158. #13063436C2×D4semidirect
213. Dic173AGL(1,ℤ/8)semidirect
224. C173#3274semidirect
238. #447D173semidirect
256. C2×D8C2×D173semidirect
268. #121C2×D173semidirect
282. #102C2×D173semidirect
284. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C2×#191497032
#260121592
D4×C22⋊D1732
#260118662
#260117432
#260117422
#260116902
#260116842
#260116382
#260116332
#260116262
#260115822
#260115662
#447×C1732
#3274173