Finite Group Databasebeta

← #46847933

Subgroups — #46847933

Order 34688.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576189C211yesmaximal, characteristic, complemented
2C2×Q16×D271C211yesmaximal, characteristic, complemented
3#31774562C211yesmaximal, characteristic, complemented
4#31774555C211yesmaximal, characteristic
5#31774152C211yesmaximal, characteristic, complemented
6#31774123C211yesmaximal, characteristic, complemented
7#31774116C211yesmaximal, characteristic, complemented
8#31774104C211yesmaximal, characteristic, complemented
9#31774056C211yesmaximal, characteristic, complemented
10#31774051C211yesmaximal, characteristic, complemented
11#31774051C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13#31774005C211yesmaximal, characteristic, complemented
14#31773970C211yesmaximal, characteristic, complemented
15#497×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19C2×Q8×D271C2211yescharacteristic
20#22576189—24no—
21C2×Dic1084C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24C2×#16864644C2211yescharacteristic
25Q16×D271—24no—
26C2×#16864642C2211yescharacteristic
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576159—21no—
30#22576146—21no—
31#22576138—21no—
32#22576138—21no—
33#22576137—21no—
34#22576136C2211yescharacteristic
35#22576136C2211yescharacteristic
36#22576136C2211yescharacteristic
37#22576133—21no—
38#22576131C2211yescharacteristic
39C4×Dic542C2211yescharacteristic
40C4×Dic542C2211yescharacteristic
41C4×D1084C2211yescharacteristic
42#22576122—21no—
43(C22⋊D271)⋊C4—21no—
44#22576092C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic
46#22576085C2211yescharacteristic
47#22576085C2211yescharacteristic
48#22576084C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576084C2211yescharacteristic
51#22576081C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic
53(C2×D271)⋊C8C2211yescharacteristic
54C2×Q16×C271C2211yescharacteristic
55C2×Q16×C271C2211yescharacteristic
56C2×SD16×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic
58C4⋊Q8×C271C2211yescharacteristic
59#124×C271C2211yescharacteristic
60#97×C271C2211yescharacteristic
61#16864653—22no—
62#16864653—41no—
63C2×Dic542—21no—
64C2×Dic542—21no—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68Q8×D271—22no—
69Q8×D271—22no—
70Q8×D271—22no—
71Q8×D271—41no—
72Q8×D271—41no—
73Q8×D271—41no—
74C2×D1084D411yescharacteristic
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C22×Dic271—21no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271C2311yescharacteristic
82C2×C4×D271C2311yescharacteristic
83Dic1084D414yescomplemented
84#16864644—22no—
85#16864644—22no—
86#16864644—22no—
87#16864642—22no—
88#16864637—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C8×D271—22no—
92C4⋊Dic271D411yescharacteristic
93C4⋊Dic271C2311yescharacteristic
94C4⋊Dic271C2311yescharacteristic
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100C22⋊Dic271—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4D411yescharacteristic
105(C2×D271)⋊C4D411yescharacteristic
106C4×Dic271—21no—
107C4×Dic271—21no—
108C2×Q8×C271C2311yescharacteristic
109C2×Q8×C271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271C2311yescharacteristic
112Q16×C271—22no—
113Q16×C271—41no—
114SD16×C271—22no—
115SD16×C271—41no—
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C4⋊C4×C271—21no—
119C4⋊C4×C271—21no—
120C22⋊C4×C271—21no—
121C22⋊C4×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271C2×D411yescharacteristic
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127Dic542—42no—
128Dic542—42no—
129Dic542—42no—
130Dic542C2×D412yes—
131Dic542C2×D412yes—
132D1084—22no—
133C22⋊D271—42no—
134C22⋊D271—42no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—42no—
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C4×D271—21no—
143C4×D271—21no—
144C4×D271—21no—
145C4×D271—21no—
146C4×D271—42no—
147C4×D271—42no—
148C4×D271—42no—
149C271⋊C8—21no—
150C271⋊C8—21no—
151C23×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271—21no—
155Q8×C271—21no—
156Q8×C271—21no—
157Q8×C271—21no—
158D4×C271—21no—
159D4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—21no—
167C8×C271C2×D412yes—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271C22×D411yescharacteristic
172C2×D271—21no—
173C2×D271—21no—
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—21no—
177C2×D271#13712yescomplemented
178C4×C271—21no—
179C4×C271—21no—
180C4×C271—41no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271C22×D411yescharacteristic
184C4×C271H(5,𝔽2)11yescharacteristic
185Dic271—21no—
186Dic271—21no—
187Dic271—22no—
188Dic271—22no—
189Dic271—41no—
190Dic271—41no—
191C2×C271—81no—
192C2×C271D4211yescharacteristic
193C2×C271C2×#13711yescharacteristic
194C2×C271#57811yescharacteristic
195D271—22no—
196D271—41no—
197C271#327011yescharacteristic, complemented, Sylow
198#3270—2711nomaximal, Sylow
199C2×#137—2711no—
200C22×Q16—2711no—
201D4×Q8—2711no—
202#548—2711no—
203#497D27111yescharacteristic, complemented
204#470—2711no—
205#465—2711no—
206#464—2711no—
207#462—2711no—
208#452—2711no—
209#451—2711no—
210#451—2711no—
211#448—2711no—
212#441—2711no—
213#435—2711no—
214C2×#41—2711no—
215C22×Q8—2711no—
216C22×Q8—2711no—
217C22×Q8—2711no—
218#137—5424no—
219C2×Q16—2711no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16—2711no—
223C2×Q16—5424no—
224C2×Q16C2×D27111yescharacteristic
225C2×Q16C2×D27111yescharacteristic
226C2×SD16—2711no—
227C2×SD16C2×D27111yescharacteristic
228C2×SD16C2×D27111yescharacteristic
229C2×M16—2711no—
230C22×C8—2711no—
231C4⋊Q8—5421no—
232C4⋊Q8C2×D27111yescharacteristic
233#124—5421no—
234#124C2×D27111yescharacteristic
235C22⋊Q8—2711no—
236C22⋊Q8—2711no—
237C22⋊Q8—2711no—
238C22⋊Q8—5421no—
239C22⋊Q8—5421no—
240C22⋊Q8—5421no—
241#121—2711no—
242#121—5421no—
243C4×Q8—2711no—
244C4×Q8—2711no—
245C4×D4—2711no—
246C4×D4—5421no—
247C4×D4—5421no—
248#106—2711no—
249C4⋊C8—2711no—
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103—2711no—
255#102—2711no—
256C22⋊C8—2711no—
257C22⋊C8—2711no—
258#97C2×D27111yescharacteristic
259#41—5422no—
260#41—10841no—
261C2×Q8—2711no—
262C2×Q8—2711no—
263C2×Q8—5421no—
264C2×Q8—5421no—
265C2×Q8—5421no—
266C2×Q8—5422no—
267C2×Q8—5422no—
268C2×Q8—5422no—
269C2×Q8—10841no—
270C2×Q8—10841no—
271C2×Q8—10841no—
272C2×Q8C22×D27111yescharacteristic
273C2×Q8C22×D27111yescharacteristic
274C2×Q8C22×D27111yescharacteristic
275C2×D4—2711no—
276C2×D4—5421no—
277C2×D4—5421no—
278C2×D4C22×D27111yescharacteristic
279C22×C4—2711no—
280C22×C4—2711no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4—5421no—
284C22×C4—5421no—
285Q16—22no—
286Q16—41no—
287Q16—2714no—
288Q16—5422no—
289Q16—5422no—
290Q16—5422no—
291SD16—22no—
292SD16—41no—
293SD16—5422no—
294M16—5422no—
295C2×C8—2711no—
296C2×C8—2711no—
297C2×C8—5422no—
298C2×C8C22×D27111yescharacteristic
299C2×C8C22×D27111yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—2711no—
303C4⋊C4—2711no—
304C4⋊C4—2711no—
305C4⋊C4—5421no—
306C4⋊C4—5421no—
307C4⋊C4—5421no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2711no—
313C22⋊C4—2711no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C42—5421no—
319C42—5421no—
320C42C22×D27111yescharacteristic
321C23—21no—
322C23—2711no—
323C23—2711no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—2712no—
331Q8—2712no—
332Q8—5421no—
333Q8—5421no—
334Q8—10842no—
335Q8—10842no—
336Q8—10842no—
337D4—21no—
338D4—21no—
339D4—5422no—
340D4—10842no—
341D4—10842no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—2711no—
346C2×C4—2711no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—10842no—
356C2×C4—10842no—
357C2×C4—10842no—
358C2×C4—10842no—
359C2×C4D4×D27111yescharacteristic
360C2×C4D4×D27111yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—21no—
363C8—5421no—
364C8—5421no—
365C8D4×D27112yes—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—2712no—
370C22—5421no—
371C22—5421no—
372C22—5421no—
373C22—5421no—
374C22—5421no—
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—5421no—
382C4—5421no—
383C4—5422no—
384C4—5422no—
385C4—10841no—
386C4—10841no—
387C4C2×D4×D27111yescharacteristic
388C4#2257620511yescharacteristic
389C2—81no—
390C2—5422no—
391C2—10841no—
392C2#3177453211yescharacteristic
393C2#137×D27111yescharacteristic
394C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#497×C271C2
socleC22×C271C22×D4
prime radical#31774555C2
hypercenter#497D271
hypocenterC271#3270
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
2. C2×Q16×D271C2semidirect
3. #31774562C2semidirect
5. #31774152C2semidirect
6. #31774123C2semidirect
7. #31774116C2semidirect
8. #31774104C2semidirect
9. #31774056C2semidirect
10. #31774051C2semidirect
11. #31774051C2semidirect
12. #31774034C2semidirect
13. #31774005C2semidirect
14. #31773970C2semidirect
15. #497×C271C2semidirect
83. Dic1084D4semidirect
177. C2×D271#137semidirect
197. C271#3270semidirect
203. #497D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
C2×Q16×D2712
#317745622
#317745552
#317741522
#317741232
#317741162
#317741042
#317740562
#317740512
#317740342
#317740052
#317739702
#497×C2712
#3270271