Finite Group Databasebeta

← #36356406

Subgroups — #36356406

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
2C22×#14583803C211yesmaximal, characteristic
3C2×C22⋊C4×D173C211yesmaximal, characteristic
4#26011382C211yesmaximal, characteristic, complemented
5#26011382C211yesmaximal, characteristic, complemented
6#26011380C211yesmaximal, characteristic, complemented
7#26011380C211yesmaximal, characteristic, complemented
8#26011364C211yesmaximal, characteristic, complemented
9#26011363C211yesmaximal, characteristic, complemented
10#26011353C211yesmaximal, characteristic, complemented
11#26011350C211yesmaximal, characteristic, complemented
12#26011346C211yesmaximal, characteristic
13C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
14C4×#14583803C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20#19149636—24no—
21C2×#14583805C2211yescharacteristic, complemented
22C2×#14583805C2211yescharacteristic, complemented
23C2×#14583805C2211yescharacteristic, complemented
24C2×#14583805C2211yescharacteristic, complemented
25C2×C22⋊Dic173C2211yescharacteristic
26C2×C22⋊Dic173C2211yescharacteristic
27C2×#14583803—24no—
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic
38C22⋊C4×D173C418yescomplemented
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
47C23×C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C42×C173C2211yescharacteristic, complemented
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×Dic173—22no—
64C22×Dic173—22no—
65C22×Dic173—41no—
66C22×Dic173C2311yescharacteristic
67C22×Dic173C2311yescharacteristic
68C22×Dic173C2311yescharacteristic
69C22×Dic173C2311yescharacteristic
70C22×Dic173C2311yescharacteristic
71C22×Dic173C2311yescharacteristic
72C2×C4×D173—22no—
73C2×C4×D173—22no—
74C2×C4×D173—41no—
75C2×C4×D173—41no—
76C2×C4×D173C2×C414yescomplemented
77C2×C4×D173C2×C414yescomplemented
78#14583805—22no—
79#14583805—22no—
80#14583805—41no—
81#14583805—41no—
82C22⋊Dic173—22no—
83C22⋊Dic173C2×C414yes—
84#14583803—22no—
85#14583803—22no—
86#14583803—22no—
87#14583803—22no—
88#14583803—22no—
89#14583803—41no—
90#14583803—41no—
91#14583803C2×C414yescomplemented
92#14583803C2×C414yescomplemented
93#14583803D414yes—
94C4×Dic173—22no—
95C4×Dic173—41no—
96C24×C173C2311yescharacteristic
97C22×C4×C173—22no—
98C22×C4×C173—22no—
99C22×C4×C173—41no—
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C22⋊C4×C173—22no—
107C22⋊C4×C173C2×C414yes—
108C42×C173—22no—
109C42×C173—41no—
110C22×D173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—28no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173—28no—
122C22×D173—44no—
123C22×D173—44no—
124C22×D173—44no—
125C22×D173—44no—
126C22×D173—44no—
127C22×D173C22×C412yes—
128C22×D173C22×C412yes—
129C22×D173C2×D412yes—
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×Dic173—44no—
147C2×Dic173C22×C412yes—
148C2×Dic173C22×C412yes—
149C2×Dic173C2×D412yes—
150C4×D173—28no—
151C4×D173—44no—
152C23×C173—21no—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173—21no—
156C23×C173C22×C412yes—
157C23×C173#4112yes—
158C23×C173#4112yes—
159C23×C173C2411yescharacteristic
160C2×C4×C173—21no—
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—44no—
176C2×C4×C173C22×C412yes—
177C2×C4×C173C22×C412yes—
178C2×C4×C173C2×D412yes—
179C22×C173—22no—
180C22×C173—22no—
181C22×C173—22no—
182C22×C173—22no—
183C22×C173—22no—
184C22×C173—22no—
185C22×C173#11714yescomplemented
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—416no—
203C2×D173—41no—
204C2×D173—41no—
205C2×D173—41no—
206C2×D173—41no—
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—24no—
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—42no—
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#388—1731no—
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⋊C4D17311yescharacteristic, complemented
247C25—1731no—
248C23×C4—1731no—
249C23×C4—1731no—
250C23×C4—1731no—
251C23×C4C2×D17311yescharacteristic
252#117—3464no—
253C2×C4⋊C4—1731no—
254C2×C4⋊C4—1731no—
255C2×C4⋊C4—1731no—
256C2×C4⋊C4—1731no—
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⋊C4—1731no—
271C2×C22⋊C4C2×D17311yescharacteristic
272C2×C22⋊C4C2×D17311yescharacteristic
273C2×C42—1731no—
274C2×C42—1731no—
275C2×C42C2×D17311yescharacteristic, complemented
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—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—3462no—
299C22×C4—3462no—
300C22×C4—3462no—
301C22×C4—3462no—
302C22×C4—1731no—
303C22×C4—1731no—
304C22×C4—1731no—
305C22×C4—1731no—
306C22×C4—1731no—
307C22×C4—1731no—
308C22×C4—1734no—
309C22×C4—1734no—
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—3462no—
320C4⋊C4—3462no—
321C4⋊C4—6921no—
322C4⋊C4—6921no—
323C22⋊C4—22no—
324C22⋊C4—3462no—
325C22⋊C4—3462no—
326C22⋊C4—3462no—
327C22⋊C4—3462no—
328C22⋊C4—3462no—
329C22⋊C4—3462no—
330C22⋊C4—1734no—
331C22⋊C4—1734no—
332C22⋊C4—1734no—
333C22⋊C4—1734no—
334C22⋊C4—6921no—
335C22⋊C4—6921no—
336C22⋊C4C4×D17314yes—
337C42—22no—
338C42—41no—
339C42—3462no—
340C42—6921no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—3468no—
346C23—3461no—
347C23—3461no—
348C23—3461no—
349C23—3461no—
350C23—3468no—
351C23—3461no—
352C23—3461no—
353C23—3461no—
354C23—3461no—
355C23—3461no—
356C23—3461no—
357C23—1732no—
358C23—1732no—
359C23—1732no—
360C23—1732no—
361C23—6924no—
362C23—6924no—
363C23—6924no—
364C23—6924no—
365C23—6924no—
366C23C2×C4×D17312yes—
367C23#1458382612yes—
368C23#1458382612yes—
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—44no—
386C2×C4—3468no—
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—3461no—
402C2×C4—1732no—
403C2×C4—1732no—
404C2×C4—1732no—
405C2×C4—6924no—
406C2×C4—6924no—
407C2×C4C2×C4×D17312yes—
408C2×C4C2×C4×D17312yes—
409C2×C4D4×D17312yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—3462no—
417C22—3462no—
418C22—3462no—
419C22—3462no—
420C22—3462no—
421C22—3462no—
422C22—3462no—
423C22—3462no—
424C22—3462no—
425C22—1734no—
426C22—69216no—
427C22—6921no—
428C22—6921no—
429C22—6921no—
430C22—6921no—
431C22#1914963614yescomplemented
432C22C22×C4×D17311yescharacteristic
433C22C2×D4×D17311yescharacteristic
434C22#41×D17311yescharacteristic
435C22C2×#1458382611yescharacteristic
436C22C2×#1458382611yescharacteristic
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—24no—
452C2—3464no—
453C2—6922no—
454C2C2×#1914963611yescharacteristic
455C2C4×D4×D17311yescharacteristic
456C2#2601196311yescharacteristic
457C2#2601213111yescharacteristic
458C2#2601213711yescharacteristic
459C2#2601216511yescharacteristic
460C2#2601217011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
FittingC4×C22⋊C4×C173C2
socle, cosocleC23×C173C24
prime radicalC24×D173C22
hypercenterC4×C22⋊C4D173
hypocenterC173#2342
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19149636C2semidirect
4. #26011382C2semidirect
5. #26011382C2semidirect
6. #26011380C2semidirect
7. #26011380C2semidirect
8. #26011364C2semidirect
9. #26011363C2semidirect
10. #26011353C2semidirect
11. #26011350C2semidirect
13. C22⋊C4×Dic173C2semidirect
14. C4×#14583803C2semidirect
15. C4×C22⋊C4×C173C2semidirect
21. C2×#14583805C22semidirect
22. C2×#14583805C22semidirect
23. C2×#14583805C22semidirect
24. C2×#14583805C22semidirect
38. C22⋊C4×D173C4semidirect
39. C2×C4×Dic173C22semidirect
40. C2×C4×Dic173C22semidirect
41. #19149586C22semidirect
42. #19149586C22semidirect
43. #19149586C22semidirect
44. #19149586C22semidirect
45. #19149586C22semidirect
46. #19149586C22semidirect
50. C2×C42×C173C22semidirect
51. C2×C42×C173C22semidirect
52. #95×C173C22semidirect
53. #95×C173C22semidirect
76. C2×C4×D173C2×C4semidirect
77. C2×C4×D173C2×C4semidirect
91. #14583803C2×C4semidirect
92. #14583803C2×C4semidirect
185. C22×C173#117semidirect
230. C173#2342semidirect
246. C4×C22⋊C4D173semidirect
275. C2×C42C2×D173semidirect
276. C2×C42C2×D173semidirect
283. #95C2×D173semidirect
284. #95C2×D173semidirect
431. C22#19149636semidirect

Maximal subgroups

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

subgroupindex
C2×#191496362
C22×#145838032
C2×C22⋊C4×D1732
#260113822
#260113802
#260113642
#260113632
#260113532
#260113502
#260113462
C22⋊C4×Dic1732
C4×#145838032
C4×C22⋊C4×C1732
#2342173