Finite Group Databasebeta

← #36370100

Subgroups — #36370100

Order 22144.

496 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
1#26012442C211yesmaximal, characteristic, complemented
2#26012441C211yesmaximal, characteristic
3#26012441C211yesmaximal, characteristic
4#26012441C211yesmaximal, characteristic
5#26012440C211yesmaximal, characteristic, complemented
6#26012440C211yesmaximal, characteristic, complemented
7#26012440C211yesmaximal, characteristic, complemented
8#26012438C211yesmaximal, characteristic
9C4⋊C4×#10621924C211yesmaximal, characteristic, complemented
10C4⋊C4×#10621924C211yesmaximal, characteristic, complemented
11C4×C4⋊#10621924C211yesmaximal, characteristic, complemented
12C22⋊C4×#10621924C211yesmaximal, characteristic
13C22⋊C4×#10621924C211yesmaximal, characteristic
14#26012427C211yesmaximal, characteristic, complemented
15#126×D173C211yesmaximal, characteristic
16C23×#10621924C2211yescharacteristic
17C2×C4⋊#10621924C2211yescharacteristic
18C2×C4⋊#10621924C2211yescharacteristic
19C2×C4⋊#10621924C2211yescharacteristic
20C2×C4⋊#10621924C2211yescharacteristic
21C2×C4⋊#10621924C2211yescharacteristic
22C2×C22⋊#10621924C2211yescharacteristic
23C2×C22⋊#10621924C2211yescharacteristic
24C2×C22⋊#10621924C2211yescharacteristic
25C2×C4×#10621924C2211yescharacteristic
26C2×C4×#10621924C2211yescharacteristic
27C2×C4×#10621924C2211yescharacteristic
28C2×C4×#10621924C2211yescharacteristic
29C2×C4×#10621924C2211yescharacteristic
30C2×C4×#10621924C2211yescharacteristic
31C2×C4×#10621924C2211yescharacteristic
32(C2×#10621924)⋊C4C2211yescharacteristic
33(C2×#10621924)⋊C4C2211yescharacteristic
34(C2×#10621924)⋊C4C2211yescharacteristic
35(C2×#10621924)⋊C4C2211yescharacteristic
36(C2×#10621924)⋊C4C2211yescharacteristic
37(C2×#10621924)⋊C4C2211yescharacteristic
38(C2×#10621924)⋊C4C2211yescharacteristic
39(C2×#10621924)⋊C4C2211yescharacteristic
40(C2×#10621924)⋊C4C2211yescharacteristic
41(C2×#10621924)⋊C4C2211yescharacteristic
42(C2×#10621924)⋊C4C2211yescharacteristic
43(C2×#10621924)⋊C4C2211yescharacteristic
44#19149674C411yescharacteristic, complemented
45#19149674C411yescharacteristic, complemented
46#19149674C411yescharacteristic, complemented
47#19149673C411yescharacteristic, complemented
48#19149672C411yescharacteristic, complemented
49#19149672C411yescharacteristic, complemented
50#19149672C411yescharacteristic, complemented
51C4⋊C4×D173C2211yescharacteristic
52C4⋊C4×D173C2211yescharacteristic
53C4⋊C4×D173C2211yescharacteristic
54C22⋊C4×D173C2211yescharacteristic
55C22⋊C4×D173C2211yescharacteristic
56C22⋊C4×D173C2211yescharacteristic
57C42×D173C2211yescharacteristic
58#126×C173C411yescharacteristic, complemented
59C22×#10621924—22no—
60C22×#10621924—22no—
61C22×#10621924—41no—
62C22×#10621924C2311yescharacteristic
63C22×#10621924C2311yescharacteristic
64C22×#10621924C2311yescharacteristic
65C22×#10621924C2311yescharacteristic
66C22×#10621924C2311yescharacteristic
67C22×#10621924C2311yescharacteristic
68C22×#10621924C2311yescharacteristic
69C22×#10621924C2311yescharacteristic
70C4⋊#10621924—22no—
71C4⋊#10621924—22no—
72C4⋊#10621924—22no—
73C4⋊#10621924—22no—
74C4⋊#10621924—41no—
75C22⋊#10621924—22no—
76C22⋊#10621924—22no—
77C22⋊#10621924—41no—
78C4×#10621924—22no—
79C4×#10621924—22no—
80C4×#10621924—22no—
81C4×#10621924—22no—
82C4×#10621924—41no—
83C4×#10621924—41no—
84C4×#10621924—41no—
85C23×D173C2311yescharacteristic
86C2×C4×D173C2311yescharacteristic
87C2×C4×D173C2311yescharacteristic
88C2×C4×D173C2311yescharacteristic
89C2×C4×D173C2311yescharacteristic
90C2×C4×D173C2311yescharacteristic
91C2×C4×D173C2311yescharacteristic
92C4⋊Dic173C2×C411yescharacteristic, complemented
93C4⋊Dic173C2×C411yescharacteristic, complemented
94C4⋊Dic173C2×C411yescharacteristic, complemented
95#14583805C2×C411yescharacteristic, complemented
96#14583805C2×C411yescharacteristic, complemented
97#14583805C2×C411yescharacteristic, complemented
98#14583805C2×C411yescharacteristic, complemented
99#14583805C2×C411yescharacteristic, complemented
100#14583805C2×C411yescharacteristic, complemented
101C22⋊Dic173C2×C411yescharacteristic
102C22⋊Dic173C2×C411yescharacteristic
103C22⋊Dic173C2×C411yescharacteristic
104#14583803C2×C411yescharacteristic
105#14583803C2×C411yescharacteristic
106#14583803C2×C411yescharacteristic
107#14583803C2×C411yescharacteristic
108#14583803C2×C411yescharacteristic
109#14583803C2×C411yescharacteristic
110C4×Dic173C2×C411yescharacteristic, complemented
111C4×Dic173C2×C411yescharacteristic, complemented
112C4×Dic173C2×C411yescharacteristic, complemented
113C4⋊C4×C173C2×C411yescharacteristic, complemented
114C4⋊C4×C173C2×C411yescharacteristic, complemented
115C4⋊C4×C173C2×C411yescharacteristic, complemented
116C22⋊C4×C173C2×C411yescharacteristic
117C22⋊C4×C173C2×C411yescharacteristic
118C22⋊C4×C173C2×C411yescharacteristic
119C42×C173C2×C411yescharacteristic, complemented
120C2×#10621924—21no—
121C2×#10621924—21no—
122C2×#10621924—21no—
123C2×#10621924—21no—
124C2×#10621924—21no—
125C2×#10621924—21no—
126C2×#10621924—21no—
127C2×#10621924—21no—
128C2×#10621924—21no—
129C2×#10621924—21no—
130C2×#10621924—21no—
131C2×#10621924—21no—
132C2×#10621924—21no—
133C2×#10621924—21no—
134C2×#10621924—21no—
135C2×#10621924—21no—
136C2×#10621924—21no—
137C2×#10621924—21no—
138C2×#10621924—21no—
139C2×#10621924—21no—
140C2×#10621924—44no—
141C2×#10621924#4112yes—
142C2×#10621924#4112yes—
143C2×#10621924#4112yes—
144C2×#10621924#4112yes—
145C22×D173—21no—
146C22×D173—21no—
147C22×D173—21no—
148C22×D173—21no—
149C22×D173—21no—
150C22×D173—21no—
151C22×D173C22×C411yescharacteristic
152C22×D173C2411yescharacteristic
153C2×Dic173C22×C411yescharacteristic
154C2×Dic173C22×C411yescharacteristic
155C2×Dic173C22×C411yescharacteristic
156C2×Dic173C22×C411yescharacteristic
157C2×Dic173C22×C411yescharacteristic
158C2×Dic173C22×C411yescharacteristic
159C4×D173—21no—
160C4×D173—21no—
161C4×D173—21no—
162C4×D173—21no—
163C4×D173—21no—
164C4×D173—21no—
165C4×D173—21no—
166C4×D173—21no—
167C4×D173—21no—
168C4×D173—21no—
169C4×D173—21no—
170C4×D173—21no—
171C23×C173C22×C411yescharacteristic
172C2×C4×C173C22×C411yescharacteristic
173C2×C4×C173C22×C411yescharacteristic
174C2×C4×C173C22×C411yescharacteristic
175C2×C4×C173C22×C411yescharacteristic
176C2×C4×C173C22×C411yescharacteristic
177C2×C4×C173C22×C411yescharacteristic
178C22×C173—21no—
179C22×C173—21no—
180C22×C173—21no—
181C22×C173C23×C411yescharacteristic
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—41no—
189C2×D173C2×#4111yescharacteristic
190C2×D173C2×#4111yescharacteristic
191C2×D173C2×#4111yescharacteristic
192C2×D173C2×#4111yescharacteristic
193C2×D173H(5,𝔽2)11yescharacteristic
194C2×D1732−1+411yescharacteristic
195#10621924—22no—
196#10621924—22no—
197#10621924—22no—
198#10621924—22no—
199#10621924—41no—
200#10621924—41no—
201#10621924—41no—
202#10621924—41no—
203C4×C173—21no—
204C4×C173—21no—
205C4×C173—21no—
206C4×C173—21no—
207C4×C173—21no—
208C4×C173—21no—
209Dic173—21no—
210Dic173—21no—
211Dic173—21no—
212Dic173—21no—
213Dic173—21no—
214Dic173—21no—
215C2×C173—41no—
216C2×C173C4×#4111yescharacteristic
217C2×C173C4×#4111yescharacteristic
218C2×C173#52011yescharacteristic
219D173—41no—
220D173#55111yescharacteristic
221D173#55211yescharacteristic
222D173#55311yescharacteristic
223D173#55611yescharacteristic
224C173#237311yescharacteristic, complemented, Sylow
225#2373—1731nomaximal, Sylow
226C2×#126—1731no—
227#389—1731no—
228#388—1731no—
229#388—1731no—
230#388—1731no—
231#387—1731no—
232#387—1731no—
233#387—1731no—
234#385—1731no—
235C4×C4⋊C4—1731no—
236C4×C4⋊C4—1731no—
237C4×C4⋊C4—1731no—
238C4×C22⋊C4—1731no—
239C4×C22⋊C4—1731no—
240#376—1731no—
241C23×C4—1731no—
242#126—1731no—
243#126—1731no—
244#126—1731no—
245#126—1731no—
246#126—1731no—
247#126—1731no—
248#126—1731no—
249#126#1062192411yescharacteristic, complemented
250C2×C4⋊C4—1731no—
251C2×C4⋊C4—1731no—
252C2×C4⋊C4—1731no—
253C2×C4⋊C4—1731no—
254C2×C4⋊C4—1731no—
255C2×C4⋊C4—1731no—
256C2×C4⋊C4—1731no—
257C2×C4⋊C4—1731no—
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×C42—1731no—
265C2×C42—1731no—
266C2×C42—1731no—
267C2×C42—1731no—
268C2×C42—1731no—
269C2×C42—1731no—
270C2×C42—1731no—
271C2×C42—1731no—
272#95—1731no—
273#95—1731no—
274#95—1731no—
275#95—1731no—
276#95—1731no—
277#95—1731no—
278#95—1731no—
279#95—1731no—
280#95—1731no—
281#95—1731no—
282#95—1731no—
283#95—1731no—
284C24—1731no—
285C22×C4—3462no—
286C22×C4—3462no—
287C22×C4—1731no—
288C22×C4—1731no—
289C22×C4—1731no—
290C22×C4—1731no—
291C22×C4—1731no—
292C22×C4—1731no—
293C22×C4—1731no—
294C22×C4—1731no—
295C22×C4—1731no—
296C22×C4—1731no—
297C22×C4—1731no—
298C22×C4—1731no—
299C22×C4—1731no—
300C22×C4—1731no—
301C22×C4—6921no—
302C4⋊C4—3462no—
303C4⋊C4—3462no—
304C4⋊C4—3462no—
305C4⋊C4—3462no—
306C4⋊C4—1731no—
307C4⋊C4—1731no—
308C4⋊C4—1731no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4—1731no—
312C4⋊C4—1731no—
313C4⋊C4—1731no—
314C4⋊C4—1731no—
315C4⋊C4—6921no—
316C4⋊C4C2×#1062192411yescharacteristic, complemented
317C4⋊C4C2×#1062192411yescharacteristic, complemented
318C4⋊C4C2×#1062192411yescharacteristic, complemented
319C22⋊C4—3462no—
320C22⋊C4—3462no—
321C22⋊C4—1731no—
322C22⋊C4—1731no—
323C22⋊C4—1731no—
324C22⋊C4—1731no—
325C22⋊C4—1731no—
326C22⋊C4—1731no—
327C22⋊C4—1731no—
328C22⋊C4—1731no—
329C22⋊C4—1731no—
330C22⋊C4—6921no—
331C22⋊C4C2×#1062192411yescharacteristic
332C22⋊C4C2×#1062192411yescharacteristic
333C22⋊C4C2×#1062192411yescharacteristic
334C42—3462no—
335C42—3462no—
336C42—3462no—
337C42—3462no—
338C42—1731no—
339C42—1731no—
340C42—1731no—
341C42—6921no—
342C42—6921no—
343C42—6921no—
344C42C2×#1062192411yescharacteristic, complemented
345C23—3461no—
346C23—3461no—
347C23—3461no—
348C23—3461no—
349C23—3461no—
350C23—3461no—
351C23—1731no—
352C23—1731no—
353C23C22×#1062192411yescharacteristic
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—3461no—
370C2×C4—3461no—
371C2×C4—3461no—
372C2×C4—3461no—
373C2×C4—3461no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—3461no—
378C2×C4—3461no—
379C2×C4—3461no—
380C2×C4—3461no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—3461no—
386C2×C4—1732no—
387C2×C4—1732no—
388C2×C4—1732no—
389C2×C4—1732no—
390C2×C4—1731no—
391C2×C4—1731no—
392C2×C4—1731no—
393C2×C4—1731no—
394C2×C4—1731no—
395C2×C4—1731no—
396C2×C4—6924no—
397C2×C4C22×#1062192411yescharacteristic
398C2×C4C22×#1062192411yescharacteristic
399C2×C4C22×#1062192411yescharacteristic
400C2×C4C22×#1062192411yescharacteristic
401C2×C4C22×#1062192411yescharacteristic
402C2×C4C22×#1062192411yescharacteristic
403C22—21no—
404C22—21no—
405C22—21no—
406C22—3461no—
407C22—3461no—
408C22—3461no—
409C22—1731no—
410C22—1731no—
411C22—1731no—
412C22—1731no—
413C22—1731no—
414C22—1731no—
415C22—6921no—
416C22—6921no—
417C22—6921no—
418C22—6921no—
419C22C23×#1062192411yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—3462no—
427C4—3462no—
428C4—3462no—
429C4—3462no—
430C4—3461no—
431C4—3461no—
432C4—3461no—
433C4—3461no—
434C4—3461no—
435C4—3461no—
436C4—6921no—
437C4—6921no—
438C4—6921no—
439C4—6921no—
440C2—41no—
441C2—1731no—
442C2—1731no—
443C2—1731no—
444C2—1731no—
445C2—6921no—
446C2#41×#1062192411yescharacteristic
447C2#41×#1062192411yescharacteristic
448C2#2601256011yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10621924
commutator, socleC22×C173C23×C4
Fitting#126×C173C4
prime radicalC23×D173C23
cosocleC22×D173C24
hypercenter#126#10621924
hypocenterC173#2373
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #26012442C2semidirect
5. #26012440C2semidirect
6. #26012440C2semidirect
7. #26012440C2semidirect
9. C4⋊C4×#10621924C2semidirect
10. C4⋊C4×#10621924C2semidirect
11. C4×C4⋊#10621924C2semidirect
14. #26012427C2semidirect
44. #19149674C4semidirect
45. #19149674C4semidirect
46. #19149674C4semidirect
47. #19149673C4semidirect
48. #19149672C4semidirect
49. #19149672C4semidirect
50. #19149672C4semidirect
58. #126×C173C4semidirect
92. C4⋊Dic173C2×C4semidirect
93. C4⋊Dic173C2×C4semidirect
94. C4⋊Dic173C2×C4semidirect
95. #14583805C2×C4semidirect
96. #14583805C2×C4semidirect
97. #14583805C2×C4semidirect
98. #14583805C2×C4semidirect
99. #14583805C2×C4semidirect
100. #14583805C2×C4semidirect
110. C4×Dic173C2×C4semidirect
111. C4×Dic173C2×C4semidirect
112. C4×Dic173C2×C4semidirect
113. C4⋊C4×C173C2×C4semidirect
114. C4⋊C4×C173C2×C4semidirect
115. C4⋊C4×C173C2×C4semidirect
119. C42×C173C2×C4semidirect
224. C173#2373semidirect
249. #126#10621924semidirect
316. C4⋊C4C2×#10621924semidirect
317. C4⋊C4C2×#10621924semidirect
318. C4⋊C4C2×#10621924semidirect
344. C42C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124422
#260124412
#260124402
#260124382
C4⋊C4×#106219242
C4×C4⋊#106219242
C22⋊C4×#106219242
#260124272
#126×D1732
#2373173