Finite Group Databasebeta

← #27158242

Subgroups — #27158242

Order 12416.

792 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690889C211yesmaximal, characteristic, complemented
2C22×C22⋊#87604C211yesmaximal, characteristic
3#20261685C212yesmaximal, complemented
4#20261685C212yesmaximal, complemented
5#20261682C212yesmaximal, complemented
6#20261678C211yesmaximal, characteristic, complemented
7#20261677C211yesmaximal, characteristic, complemented
8#20261676C211yesmaximal, characteristic
9C22⋊C4×#87604C212yesmaximal, complemented
10C4×C22⋊#87604C211yesmaximal, characteristic, complemented
11C2×C22⋊C4×D97C211yesmaximal, characteristic
12C23×#87604C2212yes—
13#15690889—24no—
14C2×C4⋊#87604C2212yescomplemented
15C2×C4⋊#87604C2212yescomplemented
16C2×C22⋊#87604—24no—
17C2×C22⋊#87604C2211yescharacteristic
18C2×C22⋊#87604C2212yes—
19C2×C22⋊#87604C2211yescharacteristic
20C2×C22⋊#87604C2211yescharacteristic
21C2×C22⋊#87604C2212yes—
22C2×C22⋊#87604C2212yes—
23C2×C22⋊#87604C2211yescharacteristic
24C2×C4×#87604C2212yescomplemented
25C2×C4×#87604C2212yescomplemented
26(C2×#87604)⋊C4C2212yescomplemented
27(C2×#87604)⋊C4C2212yescomplemented
28(C2×#87604)⋊C4C2212yescomplemented
29(C2×#87604)⋊C4C2212yescomplemented
30C24×D97C2211yescharacteristic
31C22×C4×D97C2211yescharacteristic
32C22×C4×D97C2211yescharacteristic
33C2×C22⋊Dic97C411yescharacteristic, complemented
34C2×#13095488C411yescharacteristic, complemented
35C2×#13095488C411yescharacteristic, complemented
36C22⋊C4×D97C411yescharacteristic, complemented
37C22⋊C4×D97C411yescharacteristic, complemented
38C22⋊C4×D97C411yescharacteristic, complemented
39C22⋊C4×D97C411yescharacteristic, complemented
40C22⋊C4×D97C2211yescharacteristic
41C22⋊C4×D97C2211yescharacteristic
42C22⋊C4×D97C2211yescharacteristic
43C22⋊C4×D97C2211yescharacteristic
44C2×C22⋊C4×C97C411yescharacteristic, complemented
45C22×#87604—24no—
46C22×#87604—24no—
47C22×#87604—42no—
48C22×#87604C2312yes—
49C22×#87604C2312yes—
50C22×#87604C2312yes—
51C22×#87604C2312yes—
52C4⋊#87604—24no—
53C4⋊#87604—42no—
54C22⋊#87604—22no—
55C22⋊#87604—22no—
56C22⋊#87604—24no—
57C22⋊#87604—22no—
58C22⋊#87604—24no—
59C22⋊#87604—42no—
60C22⋊#87604D414yes—
61C4×#87604—24no—
62C4×#87604—42no—
63C23×D97—22no—
64C23×D97—24no—
65C23×D97—22no—
66C23×D97—42no—
67C23×D97C2×C411yescharacteristic
68C23×D97C2×C411yescharacteristic
69C23×D97C2×C411yescharacteristic
70C23×D97C2311yescharacteristic
71C23×D97C2311yescharacteristic
72C23×D97C2311yescharacteristic
73C22×Dic97C2×C411yescharacteristic, complemented
74C22×Dic97C2×C411yescharacteristic, complemented
75C2×C4×D97—22no—
76C2×C4×D97—22no—
77C2×C4×D97—42no—
78C2×C4×D97C2×C411yescharacteristic, complemented
79C2×C4×D97C2×C411yescharacteristic, complemented
80C2×C4×D97C2×C411yescharacteristic, complemented
81C2×C4×D97C2×C411yescharacteristic, complemented
82C2×C4×D97C2311yescharacteristic
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2311yescharacteristic
85C2×C4×D97C2311yescharacteristic
86C22⋊Dic97C2×C411yescharacteristic
87C22⋊Dic97C2×C411yescharacteristic
88C22⋊Dic97C2×C411yescharacteristic, complemented
89C22⋊Dic97C2×C411yescharacteristic, complemented
90#13095488C2×C411yescharacteristic
91#13095488C2×C411yescharacteristic
92#13095488C2×C411yescharacteristic
93#13095488C2×C411yescharacteristic
94#13095488C2×C411yescharacteristic, complemented
95#13095488C2×C411yescharacteristic, complemented
96#13095488C2×C411yescharacteristic, complemented
97#13095488C2×C411yescharacteristic, complemented
98C24×C97C2×C411yescharacteristic
99C22×C4×C97C2×C411yescharacteristic, complemented
100C22×C4×C97C2×C411yescharacteristic, complemented
101C22⋊C4×C97C2×C411yescharacteristic
102C22⋊C4×C97C2×C411yescharacteristic
103C22⋊C4×C97C2×C411yescharacteristic, complemented
104C22⋊C4×C97C2×C411yescharacteristic, complemented
105C23×C97—22no—
106C23×C97—22no—
107C23×C97—41no—
108C23×C97C22×C411yescharacteristic
109C23×C97C22×C411yescharacteristic
110C23×C97C22×C411yescharacteristic
111C22×D97—21no—
112C22×D97—22no—
113C22×D97—21no—
114C22×D97—22no—
115C22×D97—22no—
116C22×D97—22no—
117C22×D97—22no—
118C22×D97—22no—
119C22×D97—22no—
120C22×D97—22no—
121C22×D97—22no—
122C22×D97—21no—
123C22×D97—22no—
124C22×D97—21no—
125C22×D97—22no—
126C22×D97—41no—
127C22×D97—41no—
128C22×D97—41no—
129C22×D97—48no—
130C22×D97—48no—
131C22×D97C22×C411yescharacteristic
132C22×D97C22×C411yescharacteristic
133C22×D97C22×C411yescharacteristic
134C22×D97C2×D412yes—
135C22×D97#4114yes—
136C22×D97#4112yes—
137C22×D97C2411yescharacteristic
138C2×#87604—22no—
139C2×#87604—22no—
140C2×#87604—22no—
141C2×#87604—22no—
142C2×#87604—22no—
143C2×#87604—22no—
144C2×#87604—22no—
145C2×#87604—22no—
146C2×#87604—22no—
147C2×#87604—22no—
148C2×#87604—22no—
149C2×#87604—48no—
150C2×#87604C2×D414yes—
151C2×C4×C97—22no—
152C2×C4×C97—41no—
153C2×C4×C97C22×C411yescharacteristic
154C2×C4×C97C22×C411yescharacteristic
155C2×C4×C97C22×C411yescharacteristic
156C2×C4×C97C22×C411yescharacteristic
157C2×Dic97—22no—
158C2×Dic97—41no—
159C2×Dic97C22×C411yescharacteristic
160C2×Dic97C22×C411yescharacteristic
161C2×Dic97C22×C411yescharacteristic
162C2×Dic97C22×C411yescharacteristic
163C4×D97—22no—
164C4×D97—22no—
165C4×D97—21no—
166C4×D97—21no—
167C4×D97—21no—
168C4×D97—21no—
169C4×D97—22no—
170C4×D97—22no—
171C4×D97—41no—
172C4×D97—41no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—44no—
181C22×C97#11712yescomplemented
182C22×C97C4×D412yes—
183C22×C97C23×C411yescharacteristic
184C2×D97—21no—
185C2×D97—21no—
186C2×D97—21no—
187C2×D97—21no—
188C2×D97—24no—
189C2×D97—24no—
190C2×D97—24no—
191C2×D97—24no—
192C2×D97—21no—
193C2×D97—21no—
194C2×D97—21no—
195C2×D97—44no—
196C2×D97—44no—
197C2×D97—44no—
198C2×D97—42no—
199C2×D97—42no—
200C2×D97#11712yescomplemented
201C2×D97C4×D412yes—
202C2×D97C22×D411yescharacteristic
203C2×D97C2×#4112yes—
204C2×D97C2×#4111yescharacteristic
205C2×D97H(5,𝔽2)12yes—
206#87604—24no—
207#87604—42no—
208#87604—42no—
209#87604—42no—
210C4×C97—21no—
211C4×C97—21no—
212C4×C97—21no—
213C4×C97—21no—
214Dic97—21no—
215Dic97—21no—
216Dic97—21no—
217Dic97—21no—
218C2×C97—22no—
219C2×C97—22no—
220C2×C97—41no—
221C2×C97C2×#11711yescharacteristic
222C2×C97C2×C4×D411yescharacteristic
223C2×C97#51811yescharacteristic
224D97—22no—
225D97—22no—
226D97—41no—
227D97#54612yes—
228D97#55112yes—
229#2342—971nomaximal, Sylow
230C97#234211yescharacteristic, complemented, Sylow
231C2×#117—971no—
232C22×C22⋊C4—971no—
233C22×C22⋊C4—971no—
234#388—972no—
235#388—972no—
236#385—972no—
237#381—971no—
238#380—971no—
239#379—971no—
240C4×C22⋊C4—971no—
241C4×C22⋊C4—972no—
242C25—971no—
243C23×C4—971no—
244C23×C4—971no—
245C23×C4—972no—
246#117—1944no—
247C2×C4⋊C4—972no—
248C2×C4⋊C4—972no—
249C2×C22⋊C4—971no—
250C2×C22⋊C4—971no—
251C2×C22⋊C4—971no—
252C2×C22⋊C4—971no—
253C2×C22⋊C4—971no—
254C2×C22⋊C4—971no—
255C2×C22⋊C4—971no—
256C2×C22⋊C4—971no—
257C2×C22⋊C4—971no—
258C2×C22⋊C4—971no—
259C2×C22⋊C4—971no—
260C2×C22⋊C4—971no—
261C2×C22⋊C4—972no—
262C2×C22⋊C4—971no—
263C2×C22⋊C4—972no—
264C2×C22⋊C4—971no—
265C2×C22⋊C4—971no—
266C2×C22⋊C4—972no—
267C2×C22⋊C4—1944no—
268C2×C22⋊C4#8760411yescharacteristic, complemented
269C2×C42—972no—
270C2×C42—972no—
271#95—972no—
272#95—972no—
273#95—972no—
274#95—972no—
275C24—971no—
276C24—971no—
277C24—971no—
278C24—971no—
279C24—971no—
280C24—971no—
281C24—3882no—
282C24—1942no—
283C24—1944no—
284C24—1942no—
285C24C2×#8760411yescharacteristic
286C22×C4—971no—
287C22×C4—972no—
288C22×C4—971no—
289C22×C4—971no—
290C22×C4—971no—
291C22×C4—971no—
292C22×C4—971no—
293C22×C4—971no—
294C22×C4—971no—
295C22×C4—971no—
296C22×C4—971no—
297C22×C4—972no—
298C22×C4—972no—
299C22×C4—972no—
300C22×C4—3882no—
301C22×C4—3882no—
302C22×C4—1942no—
303C22×C4—1942no—
304C22×C4—1944no—
305C22×C4—1944no—
306C22×C4C2×#8760411yescharacteristic, complemented
307C22×C4C2×#8760411yescharacteristic, complemented
308C4⋊C4—3882no—
309C4⋊C4—1944no—
310C22⋊C4—974no—
311C22⋊C4—971no—
312C22⋊C4—971no—
313C22⋊C4—971no—
314C22⋊C4—971no—
315C22⋊C4—971no—
316C22⋊C4—971no—
317C22⋊C4—971no—
318C22⋊C4—971no—
319C22⋊C4—971no—
320C22⋊C4—971no—
321C22⋊C4—971no—
322C22⋊C4—971no—
323C22⋊C4—3882no—
324C22⋊C4—1944no—
325C22⋊C4—1944no—
326C22⋊C4—1942no—
327C22⋊C4—1942no—
328C22⋊C4—1942no—
329C22⋊C4C2×#8760411yescharacteristic
330C22⋊C4C2×#8760411yescharacteristic
331C22⋊C4C2×#8760411yescharacteristic, complemented
332C22⋊C4C2×#8760411yescharacteristic, complemented
333C42—3882no—
334C42—1944no—
335C23—22no—
336C23—22no—
337C23—41no—
338C23—971no—
339C23—974no—
340C23—972no—
341C23—972no—
342C23—971no—
343C23—971no—
344C23—971no—
345C23—3881no—
346C23—3881no—
347C23—3881no—
348C23—3888no—
349C23—3888no—
350C23—1942no—
351C23—1941no—
352C23—1942no—
353C23—1942no—
354C23—1942no—
355C23—1942no—
356C23—1941no—
357C23—1942no—
358C23—1942no—
359C23—1942no—
360C23—1941no—
361C23—1942no—
362C23—1941no—
363C23—1942no—
364C23—1942no—
365C23C22×#8760411yescharacteristic
366C23C22×#8760411yescharacteristic
367C23C22×#8760411yescharacteristic
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—974no—
371C2×C4—971no—
372C2×C4—971no—
373C2×C4—971no—
374C2×C4—971no—
375C2×C4—3881no—
376C2×C4—3881no—
377C2×C4—3888no—
378C2×C4—3881no—
379C2×C4—1942no—
380C2×C4—1941no—
381C2×C4—1942no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4—1941no—
385C2×C4—1942no—
386C2×C4—1942no—
387C2×C4—1942no—
388C2×C4—1942no—
389C2×C4—1942no—
390C2×C4—1942no—
391C2×C4—1942no—
392C2×C4—1942no—
393C2×C4—1942no—
394C2×C4—1942no—
395C2×C4—1942no—
396C2×C4—1942no—
397C2×C4—1942no—
398C2×C4—1942no—
399C2×C4C22×#8760411yescharacteristic
400C2×C4C22×#8760411yescharacteristic
401C2×C4C22×#8760411yescharacteristic
402C2×C4C22×#8760411yescharacteristic
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—44no—
411C22—972no—
412C22—971no—
413C22—971no—
414C22—972no—
415C22—972no—
416C22—972no—
417C22—3884no—
418C22—3884no—
419C22—3882no—
420C22—3882no—
421C22—3884no—
422C22—1944no—
423C22—1944no—
424C22—1944no—
425C22—1944no—
426C22—1941no—
427C22—1941no—
428C22—1941no—
429C22—1941no—
430C22—1941no—
431C22—1941no—
432C22—1941no—
433C22#1569088912yescomplemented
434C22D4×#8760412yes—
435C22C23×#8760411yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—3882no—
441C4—3882no—
442C4—3882no—
443C4—1944no—
444C4—1941no—
445C4—1941no—
446C4—1941no—
447C4—1941no—
448C2—22no—
449C2—22no—
450C2—41no—
451C2—972no—
452C2—972no—
453C2—3881no—
454C2—1942no—
455C2—1942no—
456C2C2×#1569088911yescharacteristic
457C2C2×D4×#8760411yescharacteristic
458C2#2026180211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC2×C22⋊C4×C97C4
prime radicalC24×D97C22
cosocleC22×D97C24
hypercenterC2×C22⋊C4#87604
hypocenterC97#2342
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15690889C2semidirect
3. #20261685C2semidirect
4. #20261685C2semidirect
5. #20261682C2semidirect
6. #20261678C2semidirect
7. #20261677C2semidirect
9. C22⋊C4×#87604C2semidirect
10. C4×C22⋊#87604C2semidirect
14. C2×C4⋊#87604C22semidirect
15. C2×C4⋊#87604C22semidirect
24. C2×C4×#87604C22semidirect
25. C2×C4×#87604C22semidirect
26. (C2×#87604)⋊C4C22semidirect
27. (C2×#87604)⋊C4C22semidirect
28. (C2×#87604)⋊C4C22semidirect
29. (C2×#87604)⋊C4C22semidirect
33. C2×C22⋊Dic97C4semidirect
34. C2×#13095488C4semidirect
35. C2×#13095488C4semidirect
36. C22⋊C4×D97C4semidirect
37. C22⋊C4×D97C4semidirect
38. C22⋊C4×D97C4semidirect
39. C22⋊C4×D97C4semidirect
44. C2×C22⋊C4×C97C4semidirect
73. C22×Dic97C2×C4semidirect
74. C22×Dic97C2×C4semidirect
78. C2×C4×D97C2×C4semidirect
79. C2×C4×D97C2×C4semidirect
80. C2×C4×D97C2×C4semidirect
81. C2×C4×D97C2×C4semidirect
88. C22⋊Dic97C2×C4semidirect
89. C22⋊Dic97C2×C4semidirect
94. #13095488C2×C4semidirect
95. #13095488C2×C4semidirect
96. #13095488C2×C4semidirect
97. #13095488C2×C4semidirect
99. C22×C4×C97C2×C4semidirect
100. C22×C4×C97C2×C4semidirect
103. C22⋊C4×C97C2×C4semidirect
104. C22⋊C4×C97C2×C4semidirect
181. C22×C97#117semidirect
200. C2×D97#117semidirect
230. C97#2342semidirect
268. C2×C22⋊C4#87604semidirect
306. C22×C4C2×#87604semidirect
307. C22×C4C2×#87604semidirect
331. C22⋊C4C2×#87604semidirect
332. C22⋊C4C2×#87604semidirect
433. C22#15690889semidirect

Maximal subgroups

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

subgroupindex
C2×#156908892
C22×C22⋊#876042
#202616852
#202616822
#202616782
#202616772
#202616762
C22⋊C4×#876042
C4×C22⋊#876042
C2×C22⋊C4×D972
#234297