Finite Group Databasebeta

← #46845834

Subgroups — #46845834

Order 34688.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576192C211yesmaximal, characteristic, complemented
2C2×#22576189C211yesmaximal, characteristic, complemented
3#31774446C211yesmaximal, characteristic
4C4×Q8×D271C211yesmaximal, characteristic, complemented
5#31774120C211yesmaximal, characteristic, complemented
6#31774119C211yesmaximal, characteristic, complemented
7#31774116C211yesmaximal, characteristic, complemented
8#31774106C211yesmaximal, characteristic, complemented
9#31774105C211yesmaximal, characteristic, complemented
10#31774096C211yesmaximal, characteristic, complemented
11#31774095C211yesmaximal, characteristic, complemented
12#31773919C211yesmaximal, characteristic, complemented
13#31773897C211yesmaximal, characteristic, complemented
14#31773895C211yesmaximal, characteristic, complemented
15#464×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576192—24no—
20#22576189—24no—
21C2×Dic1084C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24C2×#16864643C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27C2×#16864637C2211yescharacteristic
28C2×#16864637C2211yescharacteristic
29#22576158C2211yescharacteristic
30#22576156C2211yescharacteristic
31#22576148C2211yescharacteristic
32#22576133—21no—
33#22576133—21no—
34Q8×Dic271C2211yescharacteristic
35C4×Dic542—21no—
36C4×Dic542C2211yescharacteristic
37#22576125—21no—
38#22576125C2211yescharacteristic
39(C22⋊D271)⋊C4—21no—
40(C22⋊D271)⋊C4—21no—
41C4⋊C4×D271—21no—
42C4⋊C4×D271C2211yescharacteristic
43C42×D271—21no—
44C42×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic
46Dic271⋊C8C2211yescharacteristic
47Dic271⋊C8C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576083C2211yescharacteristic
51#22576082C2211yescharacteristic
52#22576081C2211yescharacteristic
53#22576080C2211yescharacteristic
54C2×Q16×C271C2211yescharacteristic
55C2×SD16×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—22no—
63#16864653—22no—
64#16864653—22no—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68Q8×D271—22no—
69Q8×D271D414yescomplemented
70C2×D1084—21no—
71C2×D1084C2311yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C22×Dic271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271—21no—
79C2×C4×D271D411yescharacteristic
80C2×C4×D271D411yescharacteristic
81C2×C4×D271C2311yescharacteristic
82Dic1084—22no—
83#16864644—22no—
84#16864644—22no—
85#16864643—22no—
86#16864642—22no—
87#16864641—22no—
88#16864637—22no—
89#16864637—22no—
90C2×C271⋊C8C2311yescharacteristic
91C2×C271⋊C8C2311yescharacteristic
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101C4×Dic271—21no—
102C4×Dic271—21no—
103C4×Dic271—21no—
104C4×Dic271D411yescharacteristic
105C4×Dic271D411yescharacteristic
106C4×Dic271C2311yescharacteristic
107C2×Q8×C271C2311yescharacteristic
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110Q16×C271—22no—
111SD16×C271—22no—
112C2×C8×C271C2311yescharacteristic
113C2×C8×C271C2311yescharacteristic
114C4⋊C4×C271—21no—
115C4⋊C4×C271C2311yescharacteristic
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C42×C271—21no—
119C42×C271C2311yescharacteristic
120C22×D271—21no—
121C22×D271C2×D411yescharacteristic
122Dic542—21no—
123Dic542—24no—
124Dic542—21no—
125Dic542—21no—
126Dic542C2×D412yes—
127D1084—21no—
128D1084—21no—
129D1084—42no—
130C22⋊D271—42no—
131C22⋊D271—42no—
132C22⋊D271—42no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—42no—
137C2×Dic271C2×D411yescharacteristic
138C2×Dic271C2×D411yescharacteristic
139C2×Dic271C2×D411yescharacteristic
140C4×D271—24no—
141C4×D271—22no—
142C4×D271—41no—
143C4×D271—42no—
144C4×D271—42no—
145C4×D271C2×D412yes—
146C4×D271#4112yes—
147C271⋊C8—21no—
148C271⋊C8—21no—
149C23×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271C2×D412yes—
154D4×C271—21no—
155D4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271C2×D411yescharacteristic
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2411yescharacteristic
162C8×C271—21no—
163C8×C271—21no—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271C22×D411yescharacteristic
168C2×D271—21no—
169C2×D271—21no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271#12112yes—
174C4×C271—22no—
175C4×C271—21no—
176C4×C271—21no—
177C4×C271—41no—
178C4×C271—41no—
179C4×C271C22×D411yescharacteristic
180C4×C271C2×#4111yescharacteristic
181Dic271—22no—
182Dic271—21no—
183Dic271—41no—
184Dic271—41no—
185Dic271#12112yes—
186Dic271#13712yescomplemented
187C2×C271—81no—
188C2×C271C2×#12111yescharacteristic
189C2×C271C2×#13711yescharacteristic
190C2×C271#57511yescharacteristic
191D271—22no—
192D271—81no—
193C271#303711yescharacteristic, complemented, Sylow
194#3037—2711nomaximal, Sylow
195C2×#137—2711no—
196C2×#137—2711no—
197#532—2711no—
198C2×C4×Q8—2711no—
199#464—2711no—
200#464—2711no—
201#464—2711no—
202#464D27111yescharacteristic, complemented
203#462—2711no—
204#462—2711no—
205#460—2711no—
206#460—2711no—
207#423—2711no—
208#417—2711no—
209#417—2711no—
210C2×#41—2711no—
211C2×#41—2711no—
212C22×Q8—2711no—
213#137—5424no—
214#137—5424no—
215C2×Q16—2711no—
216C2×Q16—2711no—
217C2×Q16—2711no—
218C2×Q16C2×D27111yescharacteristic
219C2×SD16—2711no—
220C2×SD16—2711no—
221C2×SD16—2711no—
222C2×SD16C2×D27111yescharacteristic
223C2×M16—2711no—
224C2×M16—2711no—
225C4⋊Q8—2711no—
226#127—2711no—
227#124—2711no—
228#124C2×D27111yescharacteristic
229#121—5421no—
230#121—5421no—
231C4×Q8—2711no—
232C4×Q8—2711no—
233C4×Q8—2711no—
234C4×Q8—5421no—
235C4×Q8—5421no—
236C4×Q8C2×D27111yescharacteristic
237C4×D4—5421no—
238C4×D4—5421no—
239C2×C4⋊C4—2711no—
240C2×C4⋊C4—5421no—
241C2×C42—2711no—
242C2×C42—5421no—
243C4⋊C8—2711no—
244C4⋊C8—2711no—
245C4⋊C8—2711no—
246C4⋊C8C2×D27111yescharacteristic
247#103—2711no—
248#103—2711no—
249#103—2711no—
250#103C2×D27111yescharacteristic
251#102—2711no—
252#102—2711no—
253#102—2711no—
254#102C2×D27111yescharacteristic
255#41—5422no—
256#41—5422no—
257#41—5422no—
258#41—5422no—
259C2×Q8—2714no—
260C2×Q8—2711no—
261C2×Q8—2711no—
262C2×Q8—5421no—
263C2×Q8—5422no—
264C2×Q8C22×D27111yescharacteristic
265C2×Q8C22×D27111yescharacteristic
266C2×D4—2711no—
267C2×D4—5421no—
268C2×D4—5421no—
269C2×D4—5421no—
270C2×D4—5421no—
271C2×D4C22×D27111yescharacteristic
272C22×C4—2711no—
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—5421no—
276C22×C4—5421no—
277C22×C4—5421no—
278C22×C4—5421no—
279Q16—22no—
280Q16—5422no—
281Q16—5422no—
282Q16—5422no—
283SD16—22no—
284SD16—5422no—
285SD16—5422no—
286SD16—5422no—
287M16—5422no—
288M16—5422no—
289C2×C8—2711no—
290C2×C8—2711no—
291C2×C8C22×D27111yescharacteristic
292C2×C8C22×D27111yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—2711no—
295C4⋊C4—5421no—
296C4⋊C4—5421no—
297C4⋊C4—5421no—
298C4⋊C4—5421no—
299C4⋊C4—5421no—
300C4⋊C4—5421no—
301C4⋊C4C22×D27111yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—5421no—
305C22⋊C4—5421no—
306C42—21no—
307C42—2711no—
308C42—2711no—
309C42—2711no—
310C42—5421no—
311C42—5421no—
312C42—5421no—
313C42C22×D27111yescharacteristic
314C23—21no—
315C23—2711no—
316C23—5421no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—2712no—
321Q8—5424no—
322Q8—5421no—
323Q8—5421no—
324Q8—5421no—
325Q8D4×D27112yes—
326D4—21no—
327D4—21no—
328D4—5421no—
329D4—5421no—
330D4—10842no—
331D4—10842no—
332D4—10842no—
333D4—10842no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—2711no—
338C2×C4—2711no—
339C2×C4—2712no—
340C2×C4—2712no—
341C2×C4—2711no—
342C2×C4—5422no—
343C2×C4—5424no—
344C2×C4—5421no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—10841no—
348C2×C4—10842no—
349C2×C4—10842no—
350C2×C4—10842no—
351C2×C4D4×D27111yescharacteristic
352C2×C4D4×D27111yescharacteristic
353C2×C4C23×D27111yescharacteristic
354C8—21no—
355C8—21no—
356C8—5421no—
357C8—5421no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—2712no—
362C22—5421no—
363C22—5421no—
364C22—10841no—
365C22—10841no—
366C22—10841no—
367C22C2×D4×D27111yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—2712no—
374C4—2712no—
375C4—5422no—
376C4—5421no—
377C4—10841no—
378C4—10841no—
379C4C2×D4×D27111yescharacteristic
380C4#41×D27111yescharacteristic
381C2—81no—
382C2—5422no—
383C2—21681no—
384C2#121×D27111yescharacteristic
385C2#137×D27111yescharacteristic
386C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#464×C271C2
socleC22×C271C22×D4
prime radical#31774446C2
hypercenter#464D271
hypocenterC271#3037
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
2. C2×#22576189C2semidirect
4. C4×Q8×D271C2semidirect
5. #31774120C2semidirect
6. #31774119C2semidirect
7. #31774116C2semidirect
8. #31774106C2semidirect
9. #31774105C2semidirect
10. #31774096C2semidirect
11. #31774095C2semidirect
12. #31773919C2semidirect
13. #31773897C2semidirect
14. #31773895C2semidirect
15. #464×C271C2semidirect
69. Q8×D271D4semidirect
186. Dic271#137semidirect
193. C271#3037semidirect
202. #464D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
C2×#225761892
#317744462
C4×Q8×D2712
#317741202
#317741192
#317741162
#317741062
#317741052
#317740962
#317740952
#317739192
#317738972
#317738952
#464×C2712
#3037271