Finite Group Databasebeta

← #46846037

Subgroups — #46846037

Order 34688.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576187C211yesmaximal, characteristic, complemented
2C2×#22576166C211yesmaximal, characteristic, complemented
3#31774477C211yesmaximal, characteristic, complemented
4#31774463C211yesmaximal, characteristic, complemented
5#31774151C211yesmaximal, characteristic, complemented
6#31774148C211yesmaximal, characteristic, complemented
7#31774145C211yesmaximal, characteristic, complemented
8#31774140C211yesmaximal, characteristic, complemented
9#31774133C211yesmaximal, characteristic, complemented
10#31774132C211yesmaximal, characteristic, complemented
11#31774129C211yesmaximal, characteristic, complemented
12#31773936C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576187—24no—
20C2×#16864641C2211yescharacteristic
21C2×D2168C2211yescharacteristic
22C2×#16864639C2211yescharacteristic
23#22576166—24no—
24C2×#16864638C2211yescharacteristic, complemented
25C2×#16864637C2211yescharacteristic
26C2×#16864637C2211yescharacteristic, complemented
27C2×C8×D271C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic, complemented
29#22576156—21no—
30#22576146—21no—
31#22576138—21no—
32#22576136C2211yescharacteristic, complemented
33#22576135—21no—
34#22576135C2211yescharacteristic
35#22576135C2211yescharacteristic, complemented
36#22576134C2211yescharacteristic
37#22576134C2211yescharacteristic, complemented
38#22576133—21no—
39#22576133—21no—
40#22576131C2211yescharacteristic
41#22576130—21no—
42#22576128—21no—
43#22576118C2211yescharacteristic
44C4⋊C4×D271C2211yescharacteristic
45#22576094C2211yescharacteristic, complemented
46#22576093C2211yescharacteristic, complemented
47#22576091C2211yescharacteristic, complemented
48#22576084C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic
52#22576080C2211yescharacteristic
53#22576080C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic
55C22×C8×C271C2211yescharacteristic, complemented
56C22⋊Q8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864655—22no—
63#16864655—22no—
64#16864654—22no—
65#16864654—22no—
66C2×Dic542D411yescharacteristic
67C2×D1084—21no—
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084D411yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271D411yescharacteristic
78C2×C22⋊D271D411yescharacteristic
79D4×D271—22no—
80D4×D271—22no—
81C2×C4×D271—21no—
82C2×C4×D271—21no—
83C2×C4×D271C2311yescharacteristic
84C2×C4×D271C2311yescharacteristic
85#16864641—22no—
86D2168—22no—
87#16864639—22no—
88#16864638—22no—
89#16864637—22no—
90#16864637D414yescomplemented
91C2×C271⋊C8C2311yescharacteristic
92C2×C271⋊C8C2311yescharacteristic
93C8×D271—22no—
94C8×D271—22no—
95C4⋊Dic271C2311yescharacteristic
96C4⋊Dic271C2311yescharacteristic
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99C22⋊Dic271—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106C4×Dic271—21no—
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271—21no—
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111SD16×C271—22no—
112C2×C8×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C4⋊C4×C271—21no—
116C4⋊C4×C271C2311yescharacteristic
117C4⋊C4×C271C2311yescharacteristic
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—42no—
123C22×D271—81no—
124C22×D271—81no—
125C22×D271C2×D411yescharacteristic
126C22×D271C2×D411yescharacteristic
127Dic542—22no—
128D1084—22no—
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084—21no—
133D1084—42no—
134D1084—81no—
135D1084—81no—
136C22⋊D271—22no—
137C22⋊D271—22no—
138C22⋊D271—42no—
139C22⋊D271—81no—
140C22⋊D271—81no—
141C22⋊D271—81no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271C2×D411yescharacteristic
145C2×Dic271C2×D411yescharacteristic
146C4×D271—21no—
147C4×D271—21no—
148C4×D271—42no—
149C4×D271—81no—
150C4×D271C2×D412yes—
151C4×D271#4112yes—
152C271⋊C8—21no—
153C271⋊C8C2×D412yes—
154C23×C271—21no—
155C23×C271C2×D411yescharacteristic
156Q8×C271—21no—
157Q8×C271—21no—
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—81no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2411yescharacteristic
168C8×C271—21no—
169C8×C271C2×D412yes—
170C22×C271—21no—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271C22×D411yescharacteristic
177C2×D271—21no—
178C2×D271—21no—
179C2×D271—21no—
180C2×D271—21no—
181C2×D271—21no—
182C2×D271—41no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—84no—
189C2×D271#12112yes—
190C4×C271—21no—
191C4×C271—41no—
192C4×C271—41no—
193C4×C271—41no—
194C4×C271C22×D411yescharacteristic
195C4×C271C2×#4111yescharacteristic
196Dic271—21no—
197Dic271—41no—
198Dic271—41no—
199Dic271#12112yes—
200C2×C271—41no—
201C2×C271—81no—
202C2×C271C2×#12111yescharacteristic
203C2×C271#57611yescharacteristic
204C2×C271#57711yescharacteristic
205D271—22no—
206D271—41no—
207D271—81no—
208D271—81no—
209C271#306211yescharacteristic, complemented, Sylow
210#3062—2711nomaximal, Sylow
211C2×AGL(1,ℤ/8)—2711no—
212C2×#131—2711no—
213#537—2711no—
214#535—2711no—
215#469—2711no—
216#469—2711no—
217#468—2711no—
218#468—2711no—
219#466—2711no—
220#466—2711no—
221#465—2711no—
222#465D27111yescharacteristic, complemented
223#428—2711no—
224#416—2711no—
225C2×#102—2711no—
226C2×#41—2711no—
227C2×#41—2711no—
228C22×D4—2711no—
229AGL(1,ℤ/8)—5424no—
230C2×SD16—2711no—
231C2×SD16C2×D27111yescharacteristic
232C2×D8—2711no—
233C2×D8—2711no—
234#131—5424no—
235C2×M16—2711no—
236C2×M16—2711no—
237C2×M16—2711no—
238C22×C8—2711no—
239C22×C8—2711no—
240C22×C8C2×D27111yescharacteristic, complemented
241#127—5421no—
242#124—5421no—
243C22⋊Q8—2711no—
244C22⋊Q8—5421no—
245C22⋊Q8C2×D27111yescharacteristic, complemented
246#121—2711no—
247#121—2711no—
248#121—2711no—
249#121—2711no—
250#121—2711no—
251#121—5421no—
252#121—5421no—
253#121—5421no—
254#121C2×D27111yescharacteristic
255#120—5421no—
256#120—5421no—
257#117—2711no—
258C2×C4⋊C4—2711no—
259#107—2711no—
260#107—2711no—
261#106—2711no—
262#106C2×D27111yescharacteristic, complemented
263#103—2711no—
264#103C2×D27111yescharacteristic, complemented
265#102—2711no—
266#102—2711no—
267#102—2711no—
268#102—2711no—
269#102—2711no—
270#102C2×D27111yescharacteristic
271C24—5421no—
272#41—5422no—
273#41—5422no—
274#41—5422no—
275#41—5422no—
276C2×Q8—2711no—
277C2×Q8C22×D27111yescharacteristic
278C2×D4—21no—
279C2×D4—2711no—
280C2×D4—2711no—
281C2×D4—2711no—
282C2×D4—2711no—
283C2×D4—2711no—
284C2×D4—5421no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4—5422no—
288C2×D4—5422no—
289C2×D4—5421no—
290C2×D4—5421no—
291C2×D4—5421no—
292C2×D4—5421no—
293C2×D4C22×D27111yescharacteristic
294C22×C4—2711no—
295C22×C4—2711no—
296C22×C4—5421no—
297C22×C4—5421no—
298C22×C4C22×D27111yescharacteristic
299SD16—22no—
300SD16—5422no—
301D8—5422no—
302D8—5422no—
303M16—2714no—
304M16—5422no—
305M16—5422no—
306C2×C8—22no—
307C2×C8—2711no—
308C2×C8—2711no—
309C2×C8—5422no—
310C2×C8—5422no—
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2711no—
315C4⋊C4—2711no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C4⋊C4C22×D27111yescharacteristic
319C4⋊C4C22×D27111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C42—5421no—
330C23—21no—
331C23—2711no—
332C23—2711no—
333C23—5421no—
334C23—5421no—
335C23—10842no—
336C23—21681no—
337C23—21681no—
338C23D4×D27111yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—5422no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—5422no—
346D4—5422no—
347D4—5422no—
348D4—5421no—
349D4—5421no—
350D4—5421no—
351D4—5421no—
352D4—10842no—
353D4—10842no—
354D4—21681no—
355D4—21681no—
356D4—21681no—
357D4—21681no—
358D4—21681no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2711no—
365C2×C4—2712no—
366C2×C4—2712no—
367C2×C4—2711no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—10842no—
373C2×C4—21681no—
374C2×C4D4×D27111yescharacteristic
375C2×C4C23×D27111yescharacteristic
376C8—21no—
377C8—2712no—
378C8—5421no—
379C8D4×D27112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—2712no—
387C22—5421no—
388C22—5421no—
389C22—5421no—
390C22—5421no—
391C22—5421no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22—10841no—
396C22—10841no—
397C22—10841no—
398C22—21684no—
399C22C2×D4×D27111yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—2712no—
405C4—5421no—
406C4—10841no—
407C4—10841no—
408C4C2×D4×D27111yescharacteristic
409C4#41×D27111yescharacteristic
410C2—41no—
411C2—81no—
412C2—5422no—
413C2—10841no—
414C2—21681no—
415C2—21681no—
416C2#121×D27111yescharacteristic
417C2#3177469911yescharacteristic
418C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
hypercenter#465D271
hypocenterC271#3062
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576166C2semidirect
3. #31774477C2semidirect
4. #31774463C2semidirect
5. #31774151C2semidirect
6. #31774148C2semidirect
7. #31774145C2semidirect
8. #31774140C2semidirect
9. #31774133C2semidirect
10. #31774132C2semidirect
11. #31774129C2semidirect
12. #31773936C2semidirect
13. #31773892C2semidirect
14. #102×D271C2semidirect
15. #465×C271C2semidirect
24. C2×#16864638C22semidirect
26. C2×#16864637C22semidirect
28. C2×C8×D271C22semidirect
32. #22576136C22semidirect
35. #22576135C22semidirect
37. #22576134C22semidirect
45. #22576094C22semidirect
46. #22576093C22semidirect
47. #22576091C22semidirect
48. #22576084C22semidirect
50. #22576082C22semidirect
53. #22576080C22semidirect
55. C22×C8×C271C22semidirect
56. C22⋊Q8×C271C22semidirect
58. #106×C271C22semidirect
59. #103×C271C22semidirect
90. #16864637D4semidirect
209. C271#3062semidirect
222. #465D271semidirect
240. C22×C8C2×D271semidirect
245. C22⋊Q8C2×D271semidirect
262. #106C2×D271semidirect
264. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761662
#317744772
#317744632
#317741512
#317741482
#317741452
#317741402
#317741332
#317741322
#317741292
#317739362
#317738922
#102×D2712
#465×C2712
#3062271