Finite Group Databasebeta

← #46846556

Subgroups — #46846556

Order 34688.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774500C211yesmaximal, characteristic, complemented
2#31774380C211yesmaximal, characteristic, complemented
3#31774213C211yesmaximal, characteristic, complemented
4#31774211C211yesmaximal, characteristic, complemented
5#31774206C211yesmaximal, characteristic, complemented
6#31774195C211yesmaximal, characteristic, complemented
7#31774095C211yesmaximal, characteristic, complemented
8#31774090C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774006C211yesmaximal, characteristic, complemented
11#31773998C211yesmaximal, characteristic, complemented
12#31773982C211yesmaximal, characteristic, complemented
13D8×Dic271C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864641C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18#22576156C2211yescharacteristic
19#22576151C2211yescharacteristic
20#22576144—21no—
21#22576142—21no—
22#22576134C2211yescharacteristic
23#22576132—21no—
24#22576131—21no—
25Q8×Dic271C2211yescharacteristic
26Q8×Dic271C2211yescharacteristic
27D4×Dic271—21no—
28D4×Dic271—21no—
29D4×Dic271C2211yescharacteristic
30D4×Dic271C2211yescharacteristic
31D4×Dic271C2211yescharacteristic
32#22576122C2211yescharacteristic
33#22576119—21no—
34#22576119C2211yescharacteristic
35C2×C4⋊Dic271—21no—
36C2×C4⋊Dic271C2211yescharacteristic
37#22576094C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576091C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×D1084C2311yescharacteristic
60C2×C22⋊D271—21no—
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C22×Dic271—21no—
64C22×Dic271—21no—
65C22×Dic271—21no—
66C22×Dic271—21no—
67C2×C4×D271—21no—
68#16864641—22no—
69#16864639—22no—
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C4⋊Dic271—21no—
73C4⋊Dic271—21no—
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271D411yescharacteristic
77C4⋊Dic271D411yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82C22⋊Dic271—21no—
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85C22⋊Dic271D411yescharacteristic
86C22⋊Dic271D411yescharacteristic
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93#41×C271—22no—
94#41×C271—22no—
95C2×Q8×C271C2311yescharacteristic
96C2×D4×C271—21no—
97C2×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C22×C4×C271—21no—
101C22×C4×C271C2311yescharacteristic
102SD16×C271—22no—
103D8×C271—22no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22×D271—21no—
109D1084—21no—
110D1084—21no—
111C22⋊D271—81no—
112C22⋊D271—81no—
113C22⋊D271—81no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—81no—
118C2×Dic271—81no—
119C2×Dic271—81no—
120C2×Dic271—81no—
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—81no—
126C271⋊C8—21no—
127C271⋊C8—21no—
128C23×C271—21no—
129C23×C271—21no—
130C23×C271C2×D411yescharacteristic
131Q8×C271—21no—
132Q8×C271#4112yes—
133D4×C271—24no—
134D4×C271—21no—
135D4×C271—21no—
136D4×C271—21no—
137D4×C271—81no—
138D4×C271#4112yes—
139C2×C4×C271—24no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271—21no—
148C22×C271—21no—
149C22×C271—22no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271C22×D411yescharacteristic
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C4×C271—21no—
162C4×C271—22no—
163C4×C271—41no—
164C4×C271C2×#4111yescharacteristic
165C4×C271C2×#4111yescharacteristic
166Dic271—21no—
167Dic271—21no—
168Dic271—21no—
169Dic271—21no—
170Dic271—41no—
171Dic271—41no—
172Dic271—41no—
173C2×C271—41no—
174C2×C271—42no—
175C2×C271—81no—
176C2×C271#52511yescharacteristic
177C2×C271#57611yescharacteristic
178C2×C271#57711yescharacteristic
179D271—81no—
180C271#311911yescharacteristic, complemented, Sylow
181#3119—2711nomaximal, Sylow
182#540—2711no—
183#520—2711no—
184#481—2711no—
185#480—2711no—
186#479—2711no—
187#478—2711no—
188#460—2711no—
189#459—2711no—
190#449—2711no—
191#449D27111yescharacteristic, complemented
192#442—2711no—
193#440—2711no—
194C4×SD16—2711no—
195C4×D8—2711no—
196#433—2711no—
197C2×#41C2×D27111yescharacteristic
198C2×SD16—2711no—
199C2×SD16C2×D27111yescharacteristic
200C2×D8—2711no—
201C2×D8C2×D27111yescharacteristic
202#127—2711no—
203#125—2711no—
204#123—5421no—
205#123—5421no—
206#121—2711no—
207#121—5421no—
208#121—5421no—
209#121C2×D27111yescharacteristic
210C4×Q8—2711no—
211C4×Q8—2711no—
212C4×D4—2711no—
213C4×D4—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—5421no—
217C4×D4—5421no—
218#117—2711no—
219#117—5421no—
220C2×C4⋊C4—2711no—
221C2×C4⋊C4—5421no—
222#107—2711no—
223#107—2711no—
224#106—2711no—
225#106—2711no—
226C4⋊C8—2711no—
227C4⋊C8—2711no—
228#103—2711no—
229#103C2×D27111yescharacteristic
230#102—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102C2×D27111yescharacteristic
236C22⋊C8—2711no—
237C22⋊C8C2×D27111yescharacteristic
238#97—2711no—
239C4×C8—2711no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D27111yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—2711no—
246C2×D4—5421no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4C22×D27111yescharacteristic
250C2×D4C22×D27111yescharacteristic
251C22×C4—21no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4—5421no—
257C22×C4C22×D27111yescharacteristic
258SD16—22no—
259SD16—5422no—
260D8—22no—
261D8—5422no—
262C2×C8—2711no—
263C2×C8—2711no—
264C2×C8C22×D27111yescharacteristic
265C2×C8C22×D27111yescharacteristic
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—2711no—
270C4⋊C4—2711no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2711no—
279C22⋊C4—2711no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C42—2711no—
286C42—2711no—
287C42—5421no—
288C42—5421no—
289C23—21no—
290C23—21no—
291C23—5421no—
292C23D4×D27111yescharacteristic
293Q8—21no—
294Q8#1686465312yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—5421no—
301D4—5421no—
302D4—21681no—
303D4—21681no—
304D4—21681no—
305D4#1686465312yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—2711no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—2711no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—21681no—
319C2×C4—21681no—
320C2×C4—21681no—
321C2×C4—21681no—
322C2×C4—21681no—
323C2×C4D4×D27111yescharacteristic
324C2×C4C23×D27111yescharacteristic
325C8—21no—
326C8—21no—
327C8—5421no—
328C8—5421no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—10841no—
339C22—10841no—
340C22—10841no—
341C22C2×D4×D27111yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—5421no—
346C4—5421no—
347C4—5421no—
348C4—5421no—
349C4—10841no—
350C4—10841no—
351C4—10841no—
352C4C2×#1686465311yescharacteristic
353C4C2×#1686465311yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—21681no—
358C2#3177440311yescharacteristic
359C2#3177469911yescharacteristic
360C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3119
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31774500C2semidirect
2. #31774380C2semidirect
3. #31774213C2semidirect
4. #31774211C2semidirect
5. #31774206C2semidirect
6. #31774195C2semidirect
7. #31774095C2semidirect
8. #31774090C2semidirect
9. #31774042C2semidirect
10. #31774006C2semidirect
11. #31773998C2semidirect
12. #31773982C2semidirect
13. D8×Dic271C2semidirect
14. #31773955C2semidirect
15. #449×C271C2semidirect
180. C271#3119semidirect
191. #449D271semidirect

Maximal subgroups

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

subgroupindex
#317745002
#317743802
#317742132
#317742112
#317742062
#317741952
#317740952
#317740902
#317740422
#317740062
#317739982
#317739822
D8×Dic2712
#317739552
#449×C2712
#3119271