Finite Group Databasebeta

← #36356407

Subgroups — #36356407

Order 22144.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149636C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D173C211yesmaximal, characteristic
3C2×C22⋊C4×D173C211yesmaximal, characteristic
4#26011385C211yesmaximal, characteristic, complemented
5#26011384C211yesmaximal, characteristic, complemented
6#26011379C211yesmaximal, characteristic, complemented
7#26011365C211yesmaximal, characteristic, complemented
8#26011362C211yesmaximal, characteristic, complemented
9#26011353C211yesmaximal, characteristic, complemented
10#26011348C211yesmaximal, characteristic, complemented
11#26011346C211yesmaximal, characteristic
12C4×#14583803C211yesmaximal, characteristic, complemented
13#26011338C211yesmaximal, characteristic, complemented
14#26011338C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21#19149636—24no—
22C2×C4⋊Dic173C2211yescharacteristic, complemented
23C2×#14583805C2211yescharacteristic, complemented
24C2×#14583805C2211yescharacteristic, complemented
25C2×C22⋊Dic173C2211yescharacteristic
26C2×C22⋊Dic173C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C22⋊C4×D173—24no—
37C22⋊C4×D173C418yescomplemented
38C2×C4×Dic173C2211yescharacteristic, complemented
39C2×C4×Dic173C2211yescharacteristic, complemented
40C2×C4×Dic173C2211yescharacteristic, complemented
41#19149586C2211yescharacteristic, complemented
42#19149586C2211yescharacteristic, complemented
43#19149586C2211yescharacteristic, complemented
44#19149586C2211yescharacteristic, complemented
45#19149586C2211yescharacteristic, complemented
46#19149586C2211yescharacteristic, complemented
47C2×C4⋊C4×C173C2211yescharacteristic, complemented
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C42×C173C2211yescharacteristic, complemented
52#95×C173C2211yescharacteristic, complemented
53#95×C173C2211yescharacteristic, complemented
54C23×D173—22no—
55C23×D173—22no—
56C23×D173—22no—
57C23×D173—22no—
58C23×D173—41no—
59C23×D173—41no—
60C23×D173C2×C414yes—
61C23×D173C2311yescharacteristic
62C23×D173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C22×Dic173C2311yescharacteristic
66C22×Dic173C2311yescharacteristic
67C22×Dic173C2311yescharacteristic
68C22×Dic173C2311yescharacteristic
69C2×C4×D173—22no—
70C2×C4×D173—22no—
71C2×C4×D173—22no—
72C2×C4×D173—22no—
73C2×C4×D173—22no—
74C2×C4×D173—22no—
75C2×C4×D173—41no—
76C2×C4×D173—41no—
77C2×C4×D173—41no—
78C2×C4×D173—41no—
79C2×C4×D173C2×C414yescomplemented
80C2×C4×D173C2×C414yescomplemented
81C4⋊Dic173—41no—
82#14583805—22no—
83#14583805—22no—
84C22⋊Dic173—22no—
85C22⋊Dic173—41no—
86C22⋊Dic173C2×C414yescomplemented
87#14583803—22no—
88#14583803—22no—
89#14583803—22no—
90#14583803—22no—
91#14583803—22no—
92#14583803C2×C414yes—
93#14583803C2×C414yes—
94#14583803D414yes—
95C4×Dic173—22no—
96C4×Dic173—41no—
97C4×Dic173—41no—
98C24×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C4⋊C4×C173—41no—
106C22⋊C4×C173—22no—
107C22⋊C4×C173—41no—
108C22⋊C4×C173C2×C414yescomplemented
109C42×C173—22no—
110C22×D173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—28no—
119C22×D173—28no—
120C22×D173—44no—
121C22×D173—44no—
122C22×D173—44no—
123C22×D173—44no—
124C22×D173—44no—
125C22×D173C22×C412yes—
126C22×D173C22×C412yes—
127C22×D173C2×D412yes—
128C22×D173#4112yes—
129C22×D173#4112yes—
130C22×D173#4112yes—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—21no—
146C2×Dic173C22×C412yes—
147C2×Dic173C22×C412yes—
148C2×Dic173C2×D412yes—
149C4×D173—28no—
150C4×D173—44no—
151C4×D173—44no—
152C4×D173—44no—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173—21no—
156C23×C173—21no—
157C23×C173—21no—
158C23×C173—21no—
159C23×C173C22×C412yes—
160C23×C173C2411yescharacteristic
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—21no—
176C2×C4×C173C22×C412yes—
177C2×C4×C173C22×C412yes—
178C2×C4×C173C2×D412yes—
179C22×C173—22no—
180C22×C173—22no—
181C22×C173—22no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173C23×C411yescharacteristic
187C22×C173C22×D411yescharacteristic
188C22×C173C2×#4111yescharacteristic
189C22×C173C2×#4111yescharacteristic
190C22×C173C2×#4111yescharacteristic
191C22×C173H(5,𝔽2)11yescharacteristic
192C22×C173H(5,𝔽2)11yescharacteristic
193C2×D173—22no—
194C2×D173—22no—
195C2×D173—22no—
196C2×D173—22no—
197C2×D173—22no—
198C2×D173—22no—
199C2×D173—22no—
200C2×D173—22no—
201C2×D173—22no—
202C2×D173—22no—
203C2×D173—22no—
204C2×D173—22no—
205C2×D173—416no—
206C2×D173#11714yescomplemented
207C2×D173C4×D414yes—
208C4×C173—22no—
209C4×C173—22no—
210C4×C173—22no—
211C4×C173—41no—
212C4×C173—41no—
213C4×C173—41no—
214Dic173—22no—
215Dic173—22no—
216Dic173—22no—
217Dic173—41no—
218Dic173—41no—
219Dic173—41no—
220C2×C173—42no—
221C2×C173C2×#11711yescharacteristic
222C2×C173C2×C4×D411yescharacteristic
223C2×C173#51811yescharacteristic
224C2×C173#54611yescharacteristic
225C2×C173#54611yescharacteristic
226C2×C173#55111yescharacteristic
227C2×C173#55111yescharacteristic
228D173—24no—
229D173—24no—
230C173#234211yescharacteristic, complemented, Sylow
231#2342—1731nomaximal, Sylow
232C2×#117—1731no—
233C22×C22⋊C4—1731no—
234C22×C22⋊C4—1731no—
235#388—1731no—
236#388—1731no—
237#388—1731no—
238#388D17311yescharacteristic, complemented
239#385—1731no—
240#385—1731no—
241#381—1731no—
242#380—1731no—
243#379—1731no—
244C4×C22⋊C4—1731no—
245C4×C22⋊C4—1731no—
246C4×C22⋊C4—1731no—
247C25—1731no—
248C23×C4—1731no—
249C23×C4—1731no—
250C23×C4—1731no—
251C23×C4—1731no—
252#117—3464no—
253C2×C4⋊C4—1731no—
254C2×C4⋊C4—1731no—
255C2×C4⋊C4—1731no—
256C2×C4⋊C4C2×D17311yescharacteristic, complemented
257C2×C22⋊C4—3464no—
258C2×C22⋊C4—1731no—
259C2×C22⋊C4—1731no—
260C2×C22⋊C4—1731no—
261C2×C22⋊C4—1731no—
262C2×C22⋊C4—1731no—
263C2×C22⋊C4—1731no—
264C2×C22⋊C4—1731no—
265C2×C22⋊C4—1731no—
266C2×C22⋊C4—1731no—
267C2×C22⋊C4—1731no—
268C2×C22⋊C4—1738no—
269C2×C22⋊C4—1731no—
270C2×C22⋊C4C2×D17311yescharacteristic
271C2×C22⋊C4C2×D17311yescharacteristic
272C2×C22⋊C4C2×D17311yescharacteristic
273C2×C42—1731no—
274C2×C42—1731no—
275C2×C42—1731no—
276C2×C42C2×D17311yescharacteristic, complemented
277#95—1731no—
278#95—1731no—
279#95—1731no—
280#95—1731no—
281#95—1731no—
282#95—1731no—
283#95C2×D17311yescharacteristic, complemented
284#95C2×D17311yescharacteristic, complemented
285C24—3462no—
286C24—3462no—
287C24—3462no—
288C24—3462no—
289C24—1731no—
290C24—1731no—
291C24—1734no—
292C24—6921no—
293C24—6921no—
294C24C22×D17311yescharacteristic
295C22×C4—3462no—
296C22×C4—3462no—
297C22×C4—3462no—
298C22×C4—3462no—
299C22×C4—3462no—
300C22×C4—3462no—
301C22×C4—1731no—
302C22×C4—1731no—
303C22×C4—1731no—
304C22×C4—1731no—
305C22×C4—1731no—
306C22×C4—1731no—
307C22×C4—1734no—
308C22×C4—1734no—
309C22×C4—6921no—
310C22×C4—6921no—
311C22×C4—6921no—
312C22×C4—6921no—
313C22×C4C22×D17311yescharacteristic
314C22×C4C22×D17311yescharacteristic
315C22×C4C22×D17311yescharacteristic
316C22×C4C22×D17311yescharacteristic
317C22×C4C22×D17311yescharacteristic
318C22×C4C22×D17311yescharacteristic
319C4⋊C4—41no—
320C4⋊C4—3462no—
321C4⋊C4—3462no—
322C4⋊C4—6921no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—3462no—
326C22⋊C4—3462no—
327C22⋊C4—3462no—
328C22⋊C4—3462no—
329C22⋊C4—3462no—
330C22⋊C4—3462no—
331C22⋊C4—1734no—
332C22⋊C4—1734no—
333C22⋊C4—1734no—
334C22⋊C4—1734no—
335C22⋊C4—6921no—
336C22⋊C4C4×D17314yescomplemented
337C42—22no—
338C42—3462no—
339C42—6921no—
340C42—6921no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—3468no—
348C23—3468no—
349C23—3461no—
350C23—3461no—
351C23—3461no—
352C23—3461no—
353C23—3461no—
354C23—3461no—
355C23—3461no—
356C23—3461no—
357C23—1732no—
358C23—1732no—
359C23—1732no—
360C23—1732no—
361C23—1732no—
362C23—1732no—
363C23—6924no—
364C23—6924no—
365C23—6924no—
366C23—6924no—
367C23—6924no—
368C23C2×C4×D17312yes—
369C23C23×D17311yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—3468no—
386C2×C4—3461no—
387C2×C4—3461no—
388C2×C4—3461no—
389C2×C4—3461no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—3461no—
397C2×C4—3461no—
398C2×C4—3461no—
399C2×C4—3461no—
400C2×C4—3461no—
401C2×C4—1732no—
402C2×C4—1732no—
403C2×C4—1732no—
404C2×C4—6924no—
405C2×C4—6924no—
406C2×C4—6924no—
407C2×C4C2×C4×D17312yes—
408C2×C4C2×C4×D17312yes—
409C2×C4D4×D17312yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—3462no—
418C22—3462no—
419C22—3462no—
420C22—3462no—
421C22—3462no—
422C22—3462no—
423C22—3462no—
424C22—3462no—
425C22—3462no—
426C22—3462no—
427C22—3462no—
428C22—3462no—
429C22—1734no—
430C22—1734no—
431C22—69216no—
432C22C22×C4×D17311yescharacteristic
433C22C2×D4×D17311yescharacteristic
434C22#41×D17311yescharacteristic
435C22#41×D17311yescharacteristic
436C22#41×D17311yescharacteristic
437C22#1914972411yescharacteristic
438C22#1914972511yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—3462no—
446C4—3462no—
447C4—3462no—
448C4—6921no—
449C4—6921no—
450C4—6921no—
451C2—42no—
452C2—3464no—
453C2—3464no—
454C2#117×D17311yescharacteristic
455C2C4×D4×D17311yescharacteristic
456C2#2601196311yescharacteristic
457C2#2601213311yescharacteristic
458C2#2601213811yescharacteristic
459C2#2601216611yescharacteristic
460C2#2601217111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC24×D173C22
hypercenter#388D173
hypocenterC173#2342
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#19149636C2semidirect
4. #26011385C2semidirect
5. #26011384C2semidirect
6. #26011379C2semidirect
7. #26011365C2semidirect
8. #26011362C2semidirect
9. #26011353C2semidirect
10. #26011348C2semidirect
12. C4×#14583803C2semidirect
13. #26011338C2semidirect
14. #26011338C2semidirect
15. #388×C173C2semidirect
22. C2×C4⋊Dic173C22semidirect
23. C2×#14583805C22semidirect
24. C2×#14583805C22semidirect
37. C22⋊C4×D173C4semidirect
38. C2×C4×Dic173C22semidirect
39. C2×C4×Dic173C22semidirect
40. C2×C4×Dic173C22semidirect
41. #19149586C22semidirect
42. #19149586C22semidirect
43. #19149586C22semidirect
44. #19149586C22semidirect
45. #19149586C22semidirect
46. #19149586C22semidirect
47. C2×C4⋊C4×C173C22semidirect
51. C2×C42×C173C22semidirect
52. #95×C173C22semidirect
53. #95×C173C22semidirect
79. C2×C4×D173C2×C4semidirect
80. C2×C4×D173C2×C4semidirect
86. C22⋊Dic173C2×C4semidirect
108. C22⋊C4×C173C2×C4semidirect
206. C2×D173#117semidirect
230. C173#2342semidirect
238. #388D173semidirect
256. C2×C4⋊C4C2×D173semidirect
276. C2×C42C2×D173semidirect
283. #95C2×D173semidirect
284. #95C2×D173semidirect
336. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496362
C2×C22⋊C4×D1732
#260113852
#260113842
#260113792
#260113652
#260113622
#260113532
#260113482
#260113462
C4×#145838032
#260113382
#388×C1732
#2342173