Finite Group Databasebeta

← #46845900

Subgroups — #46845900

Order 34688.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576187C211yesmaximal, characteristic, complemented
2C2×#22576181C211yesmaximal, characteristic, complemented
3#31774460C211yesmaximal, characteristic, complemented
4#31774379C211yesmaximal, characteristic, complemented
5#31774119C211yesmaximal, characteristic, complemented
6#31774117C211yesmaximal, characteristic, complemented
7#31774113C211yesmaximal, characteristic, complemented
8#31774111C211yesmaximal, characteristic, complemented
9#31774095C211yesmaximal, characteristic, complemented
10#31774091C211yesmaximal, characteristic, complemented
11#31774089C211yesmaximal, characteristic, complemented
12#31773922C211yesmaximal, characteristic, complemented
13#31773895C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#460×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576187—24no—
20#22576181—24no—
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864641C2211yescharacteristic
24C2×D2168C2211yescharacteristic
25C2×#16864639C2211yescharacteristic
26C2×#16864639C2211yescharacteristic, complemented
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576156C2211yescharacteristic
30#22576148C2211yescharacteristic
31#22576147C2211yescharacteristic
32#22576132—21no—
33#22576132—21no—
34#22576128—21no—
35#22576128—21no—
36Q8×Dic271C2211yescharacteristic, complemented
37C4×D1084—21no—
38C4×D1084C2211yescharacteristic, complemented
39#22576122—21no—
40#22576122C2211yescharacteristic, complemented
41#22576117—21no—
42#22576117C2211yescharacteristic
43C4⋊C4×D271—21no—
44C4⋊C4×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46Dic271⋊C8C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic
51#22576081C2211yescharacteristic
52#22576081C2211yescharacteristic, complemented
53#22576080C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C2×SD16×C271C2211yescharacteristic, complemented
56#127×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864655—22no—
63#16864655D414yescomplemented
64#16864653—22no—
65#16864653—22no—
66C2×Dic542C2311yescharacteristic
67C2×D1084—21no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74D4×D271—22no—
75D4×D271—22no—
76C22×Dic271—21no—
77C2×C4×D271—21no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271C2311yescharacteristic
82#16864644—22no—
83#16864643—22no—
84#16864641—22no—
85D2168—22no—
86#16864639—22no—
87#16864639—22no—
88#16864637—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C8×D271—22no—
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94Dic271⋊C4—21no—
95Dic271⋊C4D411yescharacteristic
96Dic271⋊C4D411yescharacteristic
97C22⋊Dic271—21no—
98C22⋊Dic271—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4D411yescharacteristic
104(C2×D271)⋊C4D411yescharacteristic
105C4×Dic271—21no—
106C4×Dic271C2311yescharacteristic
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271—21no—
109C2×D4×C271—21no—
110C2×D4×C271C2311yescharacteristic
111C2×D4×C271C2311yescharacteristic
112SD16×C271—22no—
113SD16×C271—22no—
114C2×C8×C271C2311yescharacteristic
115C2×C8×C271C2311yescharacteristic
116C4⋊C4×C271—21no—
117C4⋊C4×C271C2311yescharacteristic
118C42×C271—21no—
119C42×C271C2311yescharacteristic
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—42no—
123C22×D271—81no—
124C22×D271—81no—
125C22×D271C2×D411yescharacteristic
126Dic542—21no—
127Dic542—21no—
128D1084—24no—
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084C2×D412yes—
133C22⋊D271—42no—
134C22⋊D271—42no—
135C22⋊D271—81no—
136C22⋊D271—81no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—42no—
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C2×Dic271C2×D411yescharacteristic
143C4×D271—24no—
144C4×D271—81no—
145C4×D271—81no—
146C4×D271—81no—
147C4×D271C2×D412yes—
148C4×D271#4112yes—
149C271⋊C8—21no—
150C271⋊C8—21no—
151C23×C271—21no—
152C23×C271—21no—
153Q8×C271—21no—
154Q8×C271C2×D412yes—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—21no—
158D4×C271—21no—
159D4×C271—81no—
160D4×C271—81no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—21no—
167C8×C271—21no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271C22×D411yescharacteristic
175C2×D271—21no—
176C2×D271—21no—
177C2×D271—22no—
178C2×D271—41no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—84no—
184C2×D271#12112yes—
185C4×C271—22no—
186C4×C271—21no—
187C4×C271—21no—
188C4×C271—41no—
189C4×C271C22×D411yescharacteristic
190C4×C271C2×#4111yescharacteristic
191Dic271—21no—
192Dic271—21no—
193Dic271—41no—
194Dic271—41no—
195Dic271#12112yes—
196C2×C271—81no—
197C2×C271—81no—
198C2×C271C2×#12111yescharacteristic
199C2×C271#57611yescharacteristic
200C2×C271#57711yescharacteristic
201D271—22no—
202D271—42no—
203D271—81no—
204C271#304511yescharacteristic, complemented, Sylow
205#3045—2711nomaximal, Sylow
206C2×AGL(1,ℤ/8)—2711no—
207C2×#135—2711no—
208#535—2711no—
209#520—2711no—
210#464—2711no—
211#464—2711no—
212#463—2711no—
213#463—2711no—
214#460—2711no—
215#460D27111yescharacteristic, complemented
216#459—2711no—
217#459—2711no—
218#424—2711no—
219#417—2711no—
220C2×#102—2711no—
221C2×#41—2711no—
222C2×#41—2711no—
223C22×D4—2711no—
224AGL(1,ℤ/8)—5424no—
225#135—5424no—
226C2×Q16—2711no—
227C2×SD16—2711no—
228C2×SD16—2711no—
229C2×SD16C2×D27111yescharacteristic
230C2×SD16C2×D27111yescharacteristic, complemented
231C2×D8—2711no—
232C2×D8—2711no—
233C2×D8—2711no—
234C2×M16—2711no—
235C22×C8—2711no—
236#127—2711no—
237#127C2×D27111yescharacteristic
238#124—2711no—
239#124—2711no—
240#121—5421no—
241#121—5421no—
242#120—5421no—
243#120—5421no—
244C4×Q8—2711no—
245C4×Q8C2×D27111yescharacteristic, complemented
246C4×D4—2711no—
247C4×D4—2711no—
248C4×D4—5421no—
249C4×D4—5421no—
250#117—2711no—
251#117—5421no—
252C2×C4⋊C4—2711no—
253C2×C4⋊C4—5421no—
254C4⋊C8—2711no—
255C4⋊C8—2711no—
256C4⋊C8—2711no—
257C4⋊C8C2×D27111yescharacteristic, complemented
258#103—2711no—
259#103—2711no—
260#102—2711no—
261#102—2711no—
262#102—2711no—
263#102—2711no—
264#102C2×D27111yescharacteristic
265#102C2×D27111yescharacteristic, complemented
266C24—5421no—
267#41—2714no—
268#41—5422no—
269#41—5422no—
270#41—5422no—
271C2×Q8—2711no—
272C2×Q8C22×D27111yescharacteristic
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—2711no—
276C2×D4—2711no—
277C2×D4—5421no—
278C2×D4—5422no—
279C2×D4—5422no—
280C2×D4—5421no—
281C2×D4—5421no—
282C2×D4—5421no—
283C2×D4—5421no—
284C2×D4C22×D27111yescharacteristic
285C2×D4C22×D27111yescharacteristic
286C22×C4—2711no—
287C22×C4—5421no—
288C22×C4—5421no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4—5421no—
292Q16—5422no—
293SD16—22no—
294SD16—22no—
295SD16—5422no—
296SD16—5422no—
297D8—5422no—
298D8—5422no—
299D8—5422no—
300M16—5422no—
301C2×C8—2711no—
302C2×C8—2711no—
303C2×C8—5422no—
304C2×C8C22×D27111yescharacteristic
305C2×C8C22×D27111yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—2711no—
308C4⋊C4—2711no—
309C4⋊C4—2711no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4C22×D27111yescharacteristic
313C22⋊C4—2711no—
314C22⋊C4—2711no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C42—21no—
322C42—2711no—
323C42—5421no—
324C42C22×D27111yescharacteristic
325C23—21no—
326C23—21no—
327C23—2711no—
328C23—5421no—
329C23—5421no—
330C23—10842no—
331C23—21681no—
332C23—21681no—
333Q8—21no—
334Q8—5421no—
335Q8—5421no—
336Q8D4×D27112yes—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—81no—
342D4—81no—
343D4—2712no—
344D4—5424no—
345D4—5421no—
346D4—5421no—
347D4—5421no—
348D4—10842no—
349D4—10842no—
350D4—21681no—
351D4—21681no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—2711no—
355C2×C4—2712no—
356C2×C4—2712no—
357C2×C4—2711no—
358C2×C4—2711no—
359C2×C4—5424no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—10842no—
363C2×C4—21681no—
364C2×C4—21681no—
365C2×C4—21681no—
366C2×C4D4×D27111yescharacteristic
367C2×C4D4×D27111yescharacteristic
368C2×C4C23×D27111yescharacteristic
369C8—21no—
370C8—21no—
371C8—5421no—
372C8—5421no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—2712no—
380C22—5421no—
381C22—5421no—
382C22—5422no—
383C22—10841no—
384C22—10841no—
385C22—10841no—
386C22—10841no—
387C22—10841no—
388C22—21684no—
389C22C2×D4×D27111yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—2712no—
395C4—5421no—
396C4—5421no—
397C4—10841no—
398C4—10841no—
399C4C2×D4×D27111yescharacteristic
400C4#41×D27111yescharacteristic
401C2—81no—
402C2—81no—
403C2—5422no—
404C2—10842no—
405C2—21681no—
406C2#121×D27111yescharacteristic
407C2#3177470111yescharacteristic
408C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#460×C271C2
socleC22×C271C22×D4
hypercenter#460D271
hypocenterC271#3045
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576181C2semidirect
3. #31774460C2semidirect
4. #31774379C2semidirect
5. #31774119C2semidirect
6. #31774117C2semidirect
7. #31774113C2semidirect
8. #31774111C2semidirect
9. #31774095C2semidirect
10. #31774091C2semidirect
11. #31774089C2semidirect
12. #31773922C2semidirect
13. #31773895C2semidirect
14. #102×D271C2semidirect
15. #460×C271C2semidirect
21. C2×#16864644C22semidirect
22. C2×#16864643C22semidirect
26. C2×#16864639C22semidirect
36. Q8×Dic271C22semidirect
38. C4×D1084C22semidirect
40. #22576122C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. Dic271⋊C8C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
49. #22576083C22semidirect
52. #22576081C22semidirect
55. C2×SD16×C271C22semidirect
57. C4×Q8×C271C22semidirect
58. C4⋊C8×C271C22semidirect
60. #102×C271C22semidirect
63. #16864655D4semidirect
204. C271#3045semidirect
215. #460D271semidirect
230. C2×SD16C2×D271semidirect
245. C4×Q8C2×D271semidirect
257. C4⋊C8C2×D271semidirect
265. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761812
#317744602
#317743792
#317741192
#317741172
#317741132
#317741112
#317740952
#317740912
#317740892
#317739222
#317738952
#102×D2712
#460×C2712
#3045271