Finite Group Databasebeta

← #46846484

Subgroups — #46846484

Order 34688.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774444C211yesmaximal, characteristic
2C4×Q8×D271C211yesmaximal, characteristic, complemented
3#31774212C211yesmaximal, characteristic, complemented
4#31774209C211yesmaximal, characteristic, complemented
5#31774191C211yesmaximal, characteristic, complemented
6#31774190C211yesmaximal, characteristic, complemented
7#31774096C211yesmaximal, characteristic, complemented
8#31774095C211yesmaximal, characteristic, complemented
9#31774037C211yesmaximal, characteristic, complemented
10#31774034C211yesmaximal, characteristic, complemented
11#31773999C211yesmaximal, characteristic, complemented
12#31773997C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773952C211yesmaximal, characteristic, complemented
15#440×C271C211yesmaximal, characteristic, complemented
16C2×#16864654—21no—
17C2×Q8×D271C2211yescharacteristic
18C2×#16864643C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864641C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576156C2211yescharacteristic
23#22576146—21no—
24#22576135C2211yescharacteristic
25#22576133—21no—
26#22576131C2211yescharacteristic
27Q8×Dic271C2211yescharacteristic
28C4×Dic542—21no—
29C4×Dic542C2211yescharacteristic
30#22576125—21no—
31#22576125C2211yescharacteristic
32C4×D1084C2211yescharacteristic
33C4×C22⋊D271—21no—
34C4⋊C4×D271—21no—
35C4⋊C4×D271C2211yescharacteristic
36C42×D271—21no—
37C42×D271C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576093C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576071C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C4×Q8×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55#107×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic
59#16864654—44no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63Q8×D271—22no—
64Q8×D271—22no—
65C2×D1084—21no—
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271D411yescharacteristic
74C2×C4×D271D411yescharacteristic
75C2×C4×D271C2311yescharacteristic
76#16864643—22no—
77#16864642—22no—
78#16864641—22no—
79C2×C271⋊C8C2311yescharacteristic
80C2×C271⋊C8C2311yescharacteristic
81C4⋊Dic271—21no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88C22⋊Dic271—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91C4×Dic271—21no—
92C4×Dic271—21no—
93C4×Dic271—21no—
94C4×Dic271D411yescharacteristic
95C4×Dic271D411yescharacteristic
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271—21no—
100SD16×C271—22no—
101C2×C8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C4⋊C4×C271—21no—
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C42×C271C2311yescharacteristic
109C22×D271—21no—
110C22×D271C2×D411yescharacteristic
111Dic542—24no—
112Dic542—21no—
113Dic542—42no—
114Dic542#4112yescomplemented
115D1084—21no—
116D1084—21no—
117D1084—42no—
118C22⋊D271—42no—
119C22⋊D271—42no—
120C22⋊D271—42no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—24no—
128C4×D271—22no—
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—22no—
132C4×D271—42no—
133C4×D271—42no—
134C271⋊C8—21no—
135C271⋊C8—21no—
136C23×C271—21no—
137Q8×C271—21no—
138Q8×C271#4112yescomplemented
139D4×C271—21no—
140D4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—42no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C8×C271—21no—
149C8×C271—21no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271C22×D411yescharacteristic
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271AGL(1,ℤ/8)12yes—
160C4×C271—22no—
161C4×C271—21no—
162C4×C271—21no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271C2×#4111yescharacteristic
166C4×C271C2×#4111yescharacteristic
167Dic271—22no—
168Dic271—21no—
169Dic271—21no—
170Dic271—41no—
171Dic271—41no—
172Dic271#13712yes—
173C2×C271—81no—
174C2×C271#52511yescharacteristic
175C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C271C2×#13711yescharacteristic
177D271—22no—
178D271—81no—
179C271#311211yescharacteristic, complemented, Sylow
180#3112—2711nomaximal, Sylow
181#532—2711no—
182C2×C4×Q8—2711no—
183#480—2711no—
184#480—2711no—
185#477—2711no—
186#477—2711no—
187#460—2711no—
188#460—2711no—
189#448—2711no—
190#448—2711no—
191#440—2711no—
192#440—2711no—
193#440—2711no—
194#440D27111yescharacteristic, complemented
195#432—2711no—
196C2×#41—5421no—
197C22×Q8—2711no—
198C2×SD16—2711no—
199C2×SD16—2711no—
200C2×SD16—2711no—
201C2×SD16C2×D27111yescharacteristic
202C4⋊Q8—2711no—
203#127—2711no—
204#124—5421no—
205#121—2711no—
206#121—2711no—
207#121—5421no—
208C4×Q8—2711no—
209C4×Q8—2711no—
210C4×Q8—2711no—
211C4×Q8—5421no—
212C4×Q8—5421no—
213C4×Q8C2×D27111yescharacteristic
214C4×D4—2711no—
215C4×D4—5421no—
216C4×D4C2×D27111yescharacteristic
217C2×C4⋊C4—2711no—
218C2×C4⋊C4—5421no—
219C2×C42—2711no—
220C2×C42—5421no—
221#107—2711no—
222#107—2711no—
223#107—2711no—
224#107C2×D27111yescharacteristic
225C4⋊C8—2711no—
226C4⋊C8—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103C2×D27111yescharacteristic
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102C2×D27111yescharacteristic
235C22⋊C8—2711no—
236C22⋊C8—2711no—
237#97—2711no—
238#97C2×D27111yescharacteristic
239#41—10844no—
240C2×Q8—2711no—
241C2×Q8—5421no—
242C2×Q8—5421no—
243C2×Q8—5422no—
244C2×Q8—5422no—
245C2×Q8C22×D27111yescharacteristic
246C2×D4—2711no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4—5421no—
250C2×D4—5421no—
251C2×D4C22×D27111yescharacteristic
252C22×C4—21no—
253C22×C4—2711no—
254C22×C4—2711no—
255C22×C4—2711no—
256C22×C4—5421no—
257C22×C4—5421no—
258C22×C4—5421no—
259SD16—22no—
260SD16—5422no—
261SD16—5422no—
262SD16—5422no—
263C2×C8—2711no—
264C2×C8—2711no—
265C2×C8C22×D27111yescharacteristic
266C2×C8C22×D27111yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—2711no—
269C4⋊C4—2711no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4C22×D27111yescharacteristic
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C42—21no—
282C42—2711no—
283C42—2711no—
284C42—2711no—
285C42—5421no—
286C42—5421no—
287C42—5421no—
288C42C22×D27111yescharacteristic
289C23—21no—
290C23—2711no—
291C23—5421no—
292Q8—21no—
293Q8—2712no—
294Q8—5424no—
295Q8—5421no—
296Q8—10842no—
297Q8#1686465412yescomplemented
298D4—21no—
299D4—21no—
300D4—5421no—
301D4—5421no—
302D4—10842no—
303D4—10842no—
304D4—10842no—
305D4—10842no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—2711no—
311C2×C4—2711no—
312C2×C4—2711no—
313C2×C4—5422no—
314C2×C4—5422no—
315C2×C4—5424no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—10842no—
322C2×C4—10842no—
323C2×C4D4×D27111yescharacteristic
324C2×C4D4×D27111yescharacteristic
325C2×C4C23×D27111yescharacteristic
326C8—21no—
327C8—21no—
328C8—5421no—
329C8—5421no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—2712no—
334C22—5421no—
335C22—5421no—
336C22—10841no—
337C22—10841no—
338C22—10841no—
339C22C2×D4×D27111yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—2712no—
346C4—5421no—
347C4—5422no—
348C4—5421no—
349C4—10841no—
350C4—10841no—
351C4#41×D27111yescharacteristic
352C4C2×#1686465411yescharacteristic
353C2—81no—
354C2—5422no—
355C2—21681no—
356C2#3177440611yescharacteristic
357C2AGL(1,ℤ/8)×D27111yescharacteristic
358C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#440×C271C2
socleC22×C271C22×D4
prime radical#31774444C2
hypercenter#440D271
hypocenterC271#3112
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D271C2semidirect
3. #31774212C2semidirect
4. #31774209C2semidirect
5. #31774191C2semidirect
6. #31774190C2semidirect
7. #31774096C2semidirect
8. #31774095C2semidirect
9. #31774037C2semidirect
10. #31774034C2semidirect
11. #31773999C2semidirect
12. #31773997C2semidirect
13. #31773995C2semidirect
14. #31773952C2semidirect
15. #440×C271C2semidirect
114. Dic542#41semidirect
138. Q8×C271#41semidirect
179. C271#3112semidirect
194. #440D271semidirect
297. Q8#16864654semidirect

Maximal subgroups

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

subgroupindex
#317744442
C4×Q8×D2712
#317742122
#317742092
#317741912
#317741902
#317740962
#317740952
#317740372
#317740342
#317739992
#317739972
#317739952
#317739522
#440×C2712
#3112271