Finite Group Databasebeta

← #2457×C211 #41010077

Subgroups — #2457×C211 #41010077

Order 27008.

596 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#123×C211C211yesmaximal, characteristic, complemented
2C2×C4×D4×C211C211yesmaximal, characteristic, complemented
3#400×C211C211yesmaximal, characteristic, complemented
4#396×C211C211yesmaximal, characteristic, complemented
5#394×C211C211yesmaximal, characteristic, complemented
6#392×C211C211yesmaximal, characteristic
7#388×C211C211yesmaximal, characteristic, complemented
8#388×C211C211yesmaximal, characteristic, complemented
9#387×C211C211yesmaximal, characteristic, complemented
10#386×C211C211yesmaximal, characteristic, complemented
11#386×C211C211yesmaximal, characteristic, complemented
12#386×C211C211yesmaximal, characteristic, complemented
13#381×C211C211yesmaximal, characteristic, complemented
14#380×C211C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C211C211yesmaximal, characteristic, complemented
16C22×D4×C211C2211yescharacteristic
17C22×D4×C211C2211yescharacteristic
18C22×D4×C211C2211yescharacteristic
19C23×C4×C211C2211yescharacteristic
20C23×C4×C211C2211yescharacteristic
21C23×C4×C211C2211yescharacteristic
22#123×C211—24no—
23C4×D4×C211—24no—
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26C2×C4⋊C4×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28C2×C4⋊C4×C211C2211yescharacteristic
29C2×C22⋊C4×C211C2211yescharacteristic
30C2×C22⋊C4×C211C2211yescharacteristic
31C2×C22⋊C4×C211C2211yescharacteristic
32C2×C22⋊C4×C211C2211yescharacteristic
33C2×C22⋊C4×C211C2211yescharacteristic
34C2×C22⋊C4×C211C2211yescharacteristic
35C2×C22⋊C4×C211C2211yescharacteristic
36C2×C22⋊C4×C211C2211yescharacteristic
37C2×C22⋊C4×C211C2211yescharacteristic
38C2×C22⋊C4×C211C2211yescharacteristic
39C2×C22⋊C4×C211C2211yescharacteristic
40C2×C22⋊C4×C211C2211yescharacteristic
41C2×C42×C211C2211yescharacteristic
42C2×C42×C211C2211yescharacteristic
43#95×C211C2211yescharacteristic
44#95×C211C2211yescharacteristic
45#95×C211C2211yescharacteristic
46#95×C211C2211yescharacteristic
47#95×C211C2211yescharacteristic
48#95×C211C2211yescharacteristic
49#95×C211C2211yescharacteristic
50#95×C211C2211yescharacteristic
51#95×C211C2211yescharacteristic
52#95×C211C2211yescharacteristic
53C24×C211C2311yescharacteristic
54C24×C211C2311yescharacteristic
55C24×C211C2311yescharacteristic
56C2×D4×C211—22no—
57C2×D4×C211—22no—
58C2×D4×C211—22no—
59C2×D4×C211—22no—
60C2×D4×C211—22no—
61C2×D4×C211—22no—
62C2×D4×C211—22no—
63C2×D4×C211—41no—
64C2×D4×C211—41no—
65C22×C4×C211—22no—
66C22×C4×C211—22no—
67C22×C4×C211—22no—
68C22×C4×C211—22no—
69C22×C4×C211—22no—
70C22×C4×C211—22no—
71C22×C4×C211—41no—
72C22×C4×C211—41no—
73C22×C4×C211—41no—
74C22×C4×C211C2311yescharacteristic
75C22×C4×C211C2311yescharacteristic
76C22×C4×C211C2311yescharacteristic
77C22×C4×C211C2311yescharacteristic
78C22×C4×C211C2311yescharacteristic
79C22×C4×C211C2311yescharacteristic
80C22×C4×C211C2311yescharacteristic
81C22×C4×C211C2311yescharacteristic
82C22×C4×C211C2311yescharacteristic
83C22×C4×C211C2311yescharacteristic
84C22×C4×C211C2311yescharacteristic
85C22×C4×C211C2311yescharacteristic
86C4⋊C4×C211—22no—
87C4⋊C4×C211—22no—
88C4⋊C4×C211—22no—
89C4⋊C4×C211—22no—
90C4⋊C4×C211—41no—
91C22⋊C4×C211—22no—
92C22⋊C4×C211—22no—
93C22⋊C4×C211—22no—
94C22⋊C4×C211—22no—
95C22⋊C4×C211—22no—
96C22⋊C4×C211—22no—
97C22⋊C4×C211—41no—
98C22⋊C4×C211—41no—
99C22⋊C4×C211—41no—
100C22⋊C4×C211—41no—
101C22⋊C4×C211—41no—
102C22⋊C4×C211D414yescomplemented
103C42×C211—22no—
104C42×C211—41no—
105C23×C211—21no—
106C23×C211—21no—
107C23×C211—21no—
108C23×C211—21no—
109C23×C211—21no—
110C23×C211—21no—
111C23×C211—21no—
112C23×C211—21no—
113C23×C211—21no—
114C23×C211—21no—
115C23×C211—21no—
116C23×C211—21no—
117C23×C211—21no—
118C23×C211—21no—
119C23×C211—21no—
120C23×C211—21no—
121C23×C211—21no—
122C23×C211—21no—
123C23×C211C2×D412yes—
124C23×C211#4112yes—
125C23×C211#4112yes—
126C23×C211C2411yescharacteristic
127D4×C211—28no—
128D4×C211—44no—
129D4×C211—44no—
130C2×C4×C211—21no—
131C2×C4×C211—21no—
132C2×C4×C211—21no—
133C2×C4×C211—21no—
134C2×C4×C211—21no—
135C2×C4×C211—21no—
136C2×C4×C211—21no—
137C2×C4×C211—21no—
138C2×C4×C211—21no—
139C2×C4×C211—21no—
140C2×C4×C211—21no—
141C2×C4×C211—21no—
142C2×C4×C211—21no—
143C2×C4×C211—21no—
144C2×C4×C211—21no—
145C2×C4×C211—21no—
146C2×C4×C211—21no—
147C2×C4×C211—21no—
148C2×C4×C211—21no—
149C2×C4×C211—21no—
150C2×C4×C211—21no—
151C2×C4×C211—21no—
152C2×C4×C211—21no—
153C2×C4×C211—21no—
154C2×C4×C211—21no—
155C2×C4×C211—21no—
156C2×C4×C211—21no—
157C2×C4×C211—21no—
158C2×C4×C211—21no—
159C2×C4×C211—21no—
160C2×C4×C211—21no—
161C2×C4×C211—21no—
162C2×C4×C211—44no—
163C2×C4×C211—44no—
164C2×C4×C211—44no—
165C2×C4×C211C2×D412yes—
166C2×C4×C211C2×D412yes—
167C2×C4×C211#4112yes—
168C2×C4×C211#4112yes—
169C22×C211—22no—
170C22×C211—22no—
171C22×C211—22no—
172C22×C211—22no—
173C22×C211—22no—
174C22×C211—22no—
175C22×C211—22no—
176C22×C211—22no—
177C22×C211—22no—
178C22×C211—41no—
179C22×C211—41no—
180C22×C211—41no—
181C22×C211—41no—
182C22×C211—41no—
183C22×C211—41no—
184C22×C211—41no—
185C22×C211—41no—
186C22×C211—41no—
187C22×C211—41no—
188C22×C211—41no—
189C22×C211—41no—
190C22×C211C22×D411yescharacteristic
191C22×C211C2×#4111yescharacteristic
192C22×C211C2×#4111yescharacteristic
193C22×C211C2×#4111yescharacteristic
194C22×C211C2×#4111yescharacteristic
195C22×C211H(5,𝔽2)11yescharacteristic
196C22×C211H(5,𝔽2)11yescharacteristic
197C4×C211—22no—
198C4×C211—22no—
199C4×C211—22no—
200C4×C211—22no—
201C4×C211—41no—
202C4×C211—41no—
203C4×C211—41no—
204C4×C211—41no—
205C4×C211—41no—
206C4×C211—41no—
207C4×C211—41no—
208C4×C211—41no—
209C2×C211—42no—
210C2×C211—42no—
211C2×C211—42no—
212C2×C211#52511yescharacteristic
213C2×C211#52911yescharacteristic
214C2×C211#54011yescharacteristic
215C2×C211#54611yescharacteristic
216C2×C211#54611yescharacteristic
217C2×C211#55111yescharacteristic
218C2×C211#55311yescharacteristic
219C211#245711yescharacteristic, direct product factor, Sylow
220#2457C21111yesmaximal, characteristic, direct product factor, Sylow
221C2×#123C2×C21111yescharacteristic, complemented
222C2×C4×D4C2×C21111yescharacteristic, complemented
223#400C2×C21111yescharacteristic, complemented
224#396C2×C21111yescharacteristic, complemented
225#394C2×C21111yescharacteristic, complemented
226#392C2×C21111yescharacteristic
227#388C2×C21111yescharacteristic, complemented
228#388C2×C21111yescharacteristic, complemented
229#387C2×C21111yescharacteristic, complemented
230#386C2×C21111yescharacteristic, complemented
231#386C2×C21111yescharacteristic, complemented
232#386C2×C21111yescharacteristic, complemented
233#381C2×C21111yescharacteristic, complemented
234#380C2×C21111yescharacteristic, complemented
235C4×C22⋊C4C2×C21111yescharacteristic, complemented
236C22×D4C22×C21111yescharacteristic
237C22×D4C22×C21111yescharacteristic
238C22×D4C22×C21111yescharacteristic
239C23×C4C22×C21111yescharacteristic
240C23×C4C22×C21111yescharacteristic
241C23×C4C22×C21111yescharacteristic
242#123—24no—
243C4×D4—24no—
244C2×C4⋊C4C22×C21111yescharacteristic
245C2×C4⋊C4C22×C21111yescharacteristic
246C2×C4⋊C4C22×C21111yescharacteristic
247C2×C4⋊C4C22×C21111yescharacteristic
248C2×C4⋊C4C22×C21111yescharacteristic
249C2×C22⋊C4C22×C21111yescharacteristic
250C2×C22⋊C4C22×C21111yescharacteristic
251C2×C22⋊C4C22×C21111yescharacteristic
252C2×C22⋊C4C22×C21111yescharacteristic
253C2×C22⋊C4C22×C21111yescharacteristic
254C2×C22⋊C4C22×C21111yescharacteristic
255C2×C22⋊C4C22×C21111yescharacteristic
256C2×C22⋊C4C22×C21111yescharacteristic
257C2×C22⋊C4C22×C21111yescharacteristic
258C2×C22⋊C4C22×C21111yescharacteristic
259C2×C22⋊C4C22×C21111yescharacteristic
260C2×C22⋊C4C22×C21111yescharacteristic
261C2×C42C22×C21111yescharacteristic
262C2×C42C22×C21111yescharacteristic
263#95C22×C21111yescharacteristic
264#95C22×C21111yescharacteristic
265#95C22×C21111yescharacteristic
266#95C22×C21111yescharacteristic
267#95C22×C21111yescharacteristic
268#95C22×C21111yescharacteristic
269#95C22×C21111yescharacteristic
270#95C22×C21111yescharacteristic
271#95C22×C21111yescharacteristic
272#95C22×C21111yescharacteristic
273C24C23×C21111yescharacteristic
274C24C23×C21111yescharacteristic
275C24C23×C21111yescharacteristic
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C23×C21111yescharacteristic
295C22×C4C23×C21111yescharacteristic
296C22×C4C23×C21111yescharacteristic
297C22×C4C23×C21111yescharacteristic
298C22×C4C23×C21111yescharacteristic
299C22×C4C23×C21111yescharacteristic
300C22×C4C23×C21111yescharacteristic
301C22×C4C23×C21111yescharacteristic
302C22×C4C23×C21111yescharacteristic
303C22×C4C23×C21111yescharacteristic
304C22×C4C23×C21111yescharacteristic
305C22×C4C23×C21111yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4D4×C21114yescomplemented
323C42—22no—
324C42—41no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23C2×D4×C21112yes—
344C23#41×C21112yes—
345C23#41×C21112yes—
346C23C24×C21111yescharacteristic
347D4—28no—
348D4—44no—
349D4—44no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
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—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4C2×D4×C21112yes—
386C2×C4C2×D4×C21112yes—
387C2×C4#41×C21112yes—
388C2×C4#41×C21112yes—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22C22×D4×C21111yescharacteristic
411C22C2×#41×C21111yescharacteristic
412C22C2×#41×C21111yescharacteristic
413C22C2×#41×C21111yescharacteristic
414C22C2×#41×C21111yescharacteristic
415C22H(5,𝔽2)×C21111yescharacteristic
416C22H(5,𝔽2)×C21111yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C2—42no—
430C2—42no—
431C2—42no—
432C2#525×C21111yescharacteristic
433C2#529×C21111yescharacteristic
434C2#540×C21111yescharacteristic
435C2#546×C21111yescharacteristic
436C2#546×C21111yescharacteristic
437C2#551×C21111yescharacteristic
438C2#553×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C24
commutator, Frattini, cosocleC23C24×C211
prime radical#392×C211C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#123×C211C2semidirect
2. C2×C4×D4×C211C2semidirect
3. #400×C211C2semidirect
4. #396×C211C2semidirect
5. #394×C211C2semidirect
7. #388×C211C2semidirect
8. #388×C211C2semidirect
9. #387×C211C2semidirect
10. #386×C211C2semidirect
11. #386×C211C2semidirect
12. #386×C211C2semidirect
13. #381×C211C2semidirect
14. #380×C211C2semidirect
15. C4×C22⋊C4×C211C2semidirect
102. C22⋊C4×C211D4semidirect
219. C211#2457direct (N × H)
220. #2457C211direct (N × H)
221. C2×#123C2×C211semidirect
222. C2×C4×D4C2×C211semidirect
223. #400C2×C211semidirect
224. #396C2×C211semidirect
225. #394C2×C211semidirect
227. #388C2×C211semidirect
228. #388C2×C211semidirect
229. #387C2×C211semidirect
230. #386C2×C211semidirect
231. #386C2×C211semidirect
232. #386C2×C211semidirect
233. #381C2×C211semidirect
234. #380C2×C211semidirect
235. C4×C22⋊C4C2×C211semidirect
322. C22⋊C4D4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#123×C2112
C2×C4×D4×C2112
#400×C2112
#396×C2112
#394×C2112
#392×C2112
#388×C2112
#387×C2112
#386×C2112
#381×C2112
#380×C2112
C4×C22⋊C4×C2112
#2457211