Finite Group Databasebeta

← #36370062

Subgroups — #36370062

Order 22144.

532 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012443C211yesmaximal, characteristic
2#26012442C211yesmaximal, characteristic, complemented
3#26012441C211yesmaximal, characteristic, complemented
4#26012441C211yesmaximal, characteristic, complemented
5#26012441C211yesmaximal, characteristic, complemented
6#26012441C211yesmaximal, characteristic, complemented
7#26012440C211yesmaximal, characteristic, complemented
8#26012440C211yesmaximal, characteristic, complemented
9#26012439C211yesmaximal, characteristic
10#26012438C211yesmaximal, characteristic, complemented
11#26012433C211yesmaximal, characteristic, complemented
12C4⋊C4×#10621924C211yesmaximal, characteristic, complemented
13C22⋊C4×#10621924C211yesmaximal, characteristic, complemented
14C4×C22⋊#10621924C211yesmaximal, characteristic, complemented
15#123×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×C22⋊#10621924C2211yescharacteristic
26C2×C22⋊#10621924C2211yescharacteristic
27C2×C22⋊#10621924C2211yescharacteristic
28C2×C22⋊#10621924C2211yescharacteristic
29C2×C4×#10621924C2211yescharacteristic
30C2×C4×#10621924C2211yescharacteristic
31C2×C4×#10621924C2211yescharacteristic
32C2×C4×#10621924C2211yescharacteristic
33C2×C4×#10621924C2211yescharacteristic
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
44C2×D4×D173C2211yescharacteristic
45C22×C4×D173C2211yescharacteristic
46#19149664C411yescharacteristic, complemented
47#19149664C411yescharacteristic, complemented
48#19149663C411yescharacteristic, complemented
49#19149662C411yescharacteristic, complemented
50#19149661C411yescharacteristic, complemented
51#19149661C411yescharacteristic, complemented
52#19149660C411yescharacteristic, complemented
53C4⋊C4×D173C2211yescharacteristic
54C4⋊C4×D173C2211yescharacteristic
55C22⋊C4×D173C2211yescharacteristic
56C22⋊C4×D173C2211yescharacteristic
57C22⋊C4×D173C2211yescharacteristic
58#123×C173C411yescharacteristic, complemented
59C22×#10621924—22no—
60C22×#10621924—41no—
61C22×#10621924—41no—
62C22×#10621924C2311yescharacteristic
63C22×#10621924C2311yescharacteristic
64C22×#10621924C2311yescharacteristic
65C22×#10621924C2311yescharacteristic
66C22×#10621924C2311yescharacteristic
67C22×#10621924C2311yescharacteristic
68C22×#10621924C2311yescharacteristic
69C22×#10621924C2311yescharacteristic
70C4⋊#10621924—22no—
71C4⋊#10621924—41no—
72C4⋊#10621924—41no—
73C4⋊#10621924—41no—
74C4⋊#10621924—41no—
75C22⋊#10621924—22no—
76C22⋊#10621924—22no—
77C22⋊#10621924—22no—
78C22⋊#10621924—41no—
79C22⋊#10621924—41no—
80C22⋊#10621924—41no—
81C22⋊#10621924—41no—
82C4×#10621924—22no—
83C4×#10621924—41no—
84C4×#10621924—41no—
85C4×#10621924—41no—
86C4×#10621924—41no—
87C23×D173C2311yescharacteristic
88C23×D173C2311yescharacteristic
89C2×D692C2×C411yescharacteristic
90C2×C22⋊D173C2×C411yescharacteristic
91C2×C22⋊D173C2×C411yescharacteristic
92D4×D173—41no—
93D4×D173—41no—
94C22×Dic173C2×C411yescharacteristic, complemented
95C2×C4×D173—41no—
96C2×C4×D173—41no—
97C2×C4×D173C2×C411yescharacteristic, complemented
98C2×C4×D173C2×C411yescharacteristic, complemented
99C2×C4×D173C2311yescharacteristic
100C2×C4×D173C2311yescharacteristic
101C2×C4×D173C2311yescharacteristic
102C2×C4×D173C2311yescharacteristic
103C2×C4×D173C2311yescharacteristic
104C4⋊Dic173C2×C411yescharacteristic, complemented
105C4⋊Dic173C2×C411yescharacteristic, complemented
106#14583805C2×C411yescharacteristic, complemented
107#14583805C2×C411yescharacteristic, complemented
108#14583805C2×C411yescharacteristic, complemented
109#14583805C2×C411yescharacteristic, complemented
110C22⋊Dic173C2×C411yescharacteristic
111C22⋊Dic173C2×C411yescharacteristic
112C22⋊Dic173C2×C411yescharacteristic, complemented
113#14583803C2×C411yescharacteristic
114#14583803C2×C411yescharacteristic
115#14583803C2×C411yescharacteristic
116#14583803C2×C411yescharacteristic
117#14583803C2×C411yescharacteristic, complemented
118#14583803C2×C411yescharacteristic, complemented
119C2×D4×C173C2×C411yescharacteristic
120C22×C4×C173C2×C411yescharacteristic, complemented
121C4⋊C4×C173C2×C411yescharacteristic, complemented
122C4⋊C4×C173C2×C411yescharacteristic, complemented
123C22⋊C4×C173C2×C411yescharacteristic
124C22⋊C4×C173C2×C411yescharacteristic
125C22⋊C4×C173C2×C411yescharacteristic, complemented
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—21no—
141C2×#10621924—21no—
142C2×#10621924—21no—
143C2×#10621924—21no—
144C2×#10621924—21no—
145C2×#10621924—21no—
146C2×#10621924—21no—
147C2×#10621924—21no—
148C2×#10621924—44no—
149C2×#10621924#4112yes—
150C2×#10621924#4112yes—
151C22×D173—21no—
152C22×D173—21no—
153C22×D173—21no—
154C22×D173—21no—
155C22×D173—21no—
156C22×D173—21no—
157C22×D173—21no—
158C22×D173—21no—
159C22×D173—21no—
160C22×D173—21no—
161C22×D173—21no—
162C22×D173—21no—
163C22×D173C22×C411yescharacteristic
164C22×D173C22×C411yescharacteristic
165C22×D173C2411yescharacteristic
166D692—41no—
167C22⋊D173—41no—
168C22⋊D173—41no—
169C2×Dic173—41no—
170C2×Dic173C22×C411yescharacteristic
171C2×Dic173C22×C411yescharacteristic
172C2×Dic173C22×C411yescharacteristic
173C2×Dic173C22×C411yescharacteristic
174C2×Dic173C22×C411yescharacteristic
175C4×D173—21no—
176C4×D173—21no—
177C4×D173—21no—
178C4×D173—21no—
179C4×D173—21no—
180C4×D173—21no—
181C4×D173—21no—
182C4×D173—21no—
183C4×D173—21no—
184C4×D173—21no—
185C4×D173—41no—
186C4×D173—41no—
187C23×C173C22×C411yescharacteristic
188C23×C173C22×C411yescharacteristic
189D4×C173—41no—
190C2×C4×C173—41no—
191C2×C4×C173C22×C411yescharacteristic
192C2×C4×C173C22×C411yescharacteristic
193C2×C4×C173C22×C411yescharacteristic
194C2×C4×C173C22×C411yescharacteristic
195C2×C4×C173C22×C411yescharacteristic
196C22×C173—21no—
197C22×C173—21no—
198C22×C173—21no—
199C22×C173—21no—
200C22×C173—21no—
201C22×C173—21no—
202C22×C173C23×C411yescharacteristic
203C2×D173—21no—
204C2×D173—21no—
205C2×D173—21no—
206C2×D173—21no—
207C2×D173—21no—
208C2×D173—21no—
209C2×D173—41no—
210C2×D173—41no—
211C2×D173—41no—
212C2×D173—41no—
213C2×D173—41no—
214C2×D173—41no—
215C2×D173—41no—
216C2×D173—41no—
217C2×D173C2×#4111yescharacteristic
218C2×D173C2×#4111yescharacteristic
219C2×D173H(5,𝔽2)11yescharacteristic
220C2×D173H(5,𝔽2)11yescharacteristic
221C2×D173H(5,𝔽2)11yescharacteristic
222C2×D1732−1+411yescharacteristic
223#10621924—22no—
224#10621924—22no—
225#10621924—41no—
226#10621924—41no—
227#10621924—41no—
228#10621924—41no—
229#10621924—41no—
230#10621924—41no—
231C4×C173—21no—
232C4×C173—21no—
233C4×C173—21no—
234C4×C173—21no—
235C4×C173—21no—
236Dic173—21no—
237Dic173—21no—
238Dic173—21no—
239Dic173—21no—
240Dic173—21no—
241C2×C173—41no—
242C2×C173—41no—
243C2×C173C4×#4111yescharacteristic
244C2×C173#51811yescharacteristic
245C2×C173#52011yescharacteristic
246D173—41no—
247D173—41no—
248D173#53811yescharacteristic
249D173#53911yescharacteristic
250D173#54011yescharacteristic
251D173#54211yescharacteristic
252C173#233511yescharacteristic, complemented, Sylow
253#2335—1731nomaximal, Sylow
254C2×#123—1731no—
255#390—1731no—
256#389—1731no—
257#388—1731no—
258#388—1731no—
259#388—1731no—
260#388—1731no—
261#387—1731no—
262#387—1731no—
263#386—1731no—
264#385—1731no—
265#380—1731no—
266C4×C4⋊C4—1731no—
267C4×C22⋊C4—1731no—
268C4×C22⋊C4—1731no—
269C22×D4—1731no—
270C23×C4—1731no—
271C23×C4—1731no—
272#123—1731no—
273#123—1731no—
274#123—1731no—
275#123—1731no—
276#123—1731no—
277#123—1731no—
278#123—1731no—
279#123#1062192411yescharacteristic, complemented
280C2×C4⋊C4—1731no—
281C2×C4⋊C4—1731no—
282C2×C4⋊C4—1731no—
283C2×C4⋊C4—1731no—
284C2×C4⋊C4—1731no—
285C2×C4⋊C4—1731no—
286C2×C4⋊C4—1731no—
287C2×C22⋊C4—1731no—
288C2×C22⋊C4—1731no—
289C2×C22⋊C4—1731no—
290C2×C22⋊C4—1731no—
291C2×C22⋊C4—1731no—
292C2×C22⋊C4—1731no—
293C2×C22⋊C4—1731no—
294C2×C22⋊C4—1731no—
295C2×C22⋊C4—1731no—
296C2×C22⋊C4—1731no—
297C2×C42—1731no—
298C2×C42—1731no—
299C2×C42—1731no—
300C2×C42—1731no—
301C2×C42—1731no—
302#95—1731no—
303#95—1731no—
304#95—1731no—
305#95—1731no—
306#95—1731no—
307#95—1731no—
308#95—1731no—
309#95—1731no—
310#95—1731no—
311#95—1731no—
312C24—1731no—
313C24—1731no—
314C2×D4—1731no—
315C2×D4—1731no—
316C2×D4—1731no—
317C2×D4—6921no—
318C2×D4—6921no—
319C2×D4C2×#1062192411yescharacteristic
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—1731no—
331C22×C4—1731no—
332C22×C4—1731no—
333C22×C4—1731no—
334C22×C4—1731no—
335C22×C4—1731no—
336C22×C4—1731no—
337C22×C4—6921no—
338C22×C4—6921no—
339C22×C4—6921no—
340C22×C4—6921no—
341C22×C4C2×#1062192411yescharacteristic, complemented
342C4⋊C4—3462no—
343C4⋊C4—1731no—
344C4⋊C4—1731no—
345C4⋊C4—1731no—
346C4⋊C4—1731no—
347C4⋊C4—1731no—
348C4⋊C4—1731no—
349C4⋊C4—6921no—
350C4⋊C4—6921no—
351C4⋊C4—6921no—
352C4⋊C4—6921no—
353C4⋊C4C2×#1062192411yescharacteristic, complemented
354C4⋊C4C2×#1062192411yescharacteristic, complemented
355C22⋊C4—3462no—
356C22⋊C4—3462no—
357C22⋊C4—3462no—
358C22⋊C4—1731no—
359C22⋊C4—1731no—
360C22⋊C4—1731no—
361C22⋊C4—1731no—
362C22⋊C4—1731no—
363C22⋊C4—1731no—
364C22⋊C4—1731no—
365C22⋊C4—1731no—
366C22⋊C4—1731no—
367C22⋊C4—6921no—
368C22⋊C4—6921no—
369C22⋊C4—6921no—
370C22⋊C4—6921no—
371C22⋊C4C2×#1062192411yescharacteristic
372C22⋊C4C2×#1062192411yescharacteristic
373C22⋊C4C2×#1062192411yescharacteristic, complemented
374C42—3462no—
375C42—6921no—
376C42—6921no—
377C42—6921no—
378C42—6921no—
379C23—3461no—
380C23—3461no—
381C23—3461no—
382C23—3461no—
383C23—3461no—
384C23—3461no—
385C23—3461no—
386C23—3461no—
387C23—3461no—
388C23—3461no—
389C23—3461no—
390C23—3461no—
391C23—1731no—
392C23—1731no—
393C23—1731no—
394C23C22×#1062192411yescharacteristic
395C23C22×#1062192411yescharacteristic
396D4—41no—
397D4—6921no—
398D4—6921no—
399D4—6921no—
400C2×C4—41no—
401C2×C4—3461no—
402C2×C4—3461no—
403C2×C4—3461no—
404C2×C4—3461no—
405C2×C4—3461no—
406C2×C4—3461no—
407C2×C4—3461no—
408C2×C4—3461no—
409C2×C4—3461no—
410C2×C4—3461no—
411C2×C4—3461no—
412C2×C4—3461no—
413C2×C4—3461no—
414C2×C4—3461no—
415C2×C4—3461no—
416C2×C4—3461no—
417C2×C4—3461no—
418C2×C4—3461no—
419C2×C4—3461no—
420C2×C4—3461no—
421C2×C4—3461no—
422C2×C4—3461no—
423C2×C4—3461no—
424C2×C4—3461no—
425C2×C4—3461no—
426C2×C4—3461no—
427C2×C4—3461no—
428C2×C4—3461no—
429C2×C4—3461no—
430C2×C4—3461no—
431C2×C4—3461no—
432C2×C4—3461no—
433C2×C4—1732no—
434C2×C4—1732no—
435C2×C4—1731no—
436C2×C4—1731no—
437C2×C4—1731no—
438C2×C4—1731no—
439C2×C4—1731no—
440C2×C4—6921no—
441C2×C4—6921no—
442C2×C4—6921no—
443C2×C4—6924no—
444C2×C4C22×#1062192411yescharacteristic
445C2×C4C22×#1062192411yescharacteristic
446C2×C4C22×#1062192411yescharacteristic
447C2×C4C22×#1062192411yescharacteristic
448C2×C4C22×#1062192411yescharacteristic
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—3461no—
456C22—3461no—
457C22—3461no—
458C22—3461no—
459C22—3461no—
460C22—3461no—
461C22—1731no—
462C22—1731no—
463C22—1731no—
464C22—1731no—
465C22—1731no—
466C22—1731no—
467C22—6921no—
468C22—6921no—
469C22—6921no—
470C22—6921no—
471C22—6921no—
472C22—6921no—
473C22—6921no—
474C22—6921no—
475C22C23×#1062192411yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—3462no—
482C4—3462no—
483C4—3461no—
484C4—3461no—
485C4—3461no—
486C4—3461no—
487C4—3461no—
488C4—6921no—
489C4—6921no—
490C4—6921no—
491C4—6921no—
492C4—6921no—
493C4—6921no—
494C2—41no—
495C2—41no—
496C2—1731no—
497C2—1731no—
498C2—1731no—
499C2—1731no—
500C2—6921no—
501C2—6921no—
502C2#41×#1062192411yescharacteristic
503C2#2601255811yescharacteristic
504C2#2601256011yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10621924
commutator, socleC22×C173C23×C4
Fitting#123×C173C4
prime radicalC2×D4×D173C22
cosocleC22×D173C24
hypercenter#123#10621924
hypocenterC173#2335
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #26012442C2semidirect
3. #26012441C2semidirect
4. #26012441C2semidirect
5. #26012441C2semidirect
6. #26012441C2semidirect
7. #26012440C2semidirect
8. #26012440C2semidirect
10. #26012438C2semidirect
11. #26012433C2semidirect
12. C4⋊C4×#10621924C2semidirect
13. C22⋊C4×#10621924C2semidirect
14. C4×C22⋊#10621924C2semidirect
46. #19149664C4semidirect
47. #19149664C4semidirect
48. #19149663C4semidirect
49. #19149662C4semidirect
50. #19149661C4semidirect
51. #19149661C4semidirect
52. #19149660C4semidirect
58. #123×C173C4semidirect
94. C22×Dic173C2×C4semidirect
97. C2×C4×D173C2×C4semidirect
98. C2×C4×D173C2×C4semidirect
104. C4⋊Dic173C2×C4semidirect
105. C4⋊Dic173C2×C4semidirect
106. #14583805C2×C4semidirect
107. #14583805C2×C4semidirect
108. #14583805C2×C4semidirect
109. #14583805C2×C4semidirect
112. C22⋊Dic173C2×C4semidirect
117. #14583803C2×C4semidirect
118. #14583803C2×C4semidirect
120. C22×C4×C173C2×C4semidirect
121. C4⋊C4×C173C2×C4semidirect
122. C4⋊C4×C173C2×C4semidirect
125. C22⋊C4×C173C2×C4semidirect
252. C173#2335semidirect
279. #123#10621924semidirect
341. C22×C4C2×#10621924semidirect
353. C4⋊C4C2×#10621924semidirect
354. C4⋊C4C2×#10621924semidirect
373. C22⋊C4C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124432
#260124422
#260124412
#260124402
#260124392
#260124382
#260124332
C4⋊C4×#106219242
C22⋊C4×#106219242
C4×C22⋊#106219242
#123×D1732
#2335173