Finite Group Databasebeta

← #46846021

Subgroups — #46846021

Order 34688.

484 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576181C211yesmaximal, characteristic, complemented
2C2×#22576166C211yesmaximal, characteristic, complemented
3#31774477C211yesmaximal, characteristic, complemented
4#31774476C211yesmaximal, characteristic, complemented
5#31774154C211yesmaximal, characteristic, complemented
6#31774153C211yesmaximal, characteristic, complemented
7#31774149C211yesmaximal, characteristic, complemented
8#31774148C211yesmaximal, characteristic, complemented
9#31774127C211yesmaximal, characteristic, complemented
10#31774126C211yesmaximal, characteristic, complemented
11#31774123C211yesmaximal, characteristic, complemented
12#106×D271C211yesmaximal, characteristic, complemented
13#31773897C211yesmaximal, characteristic, complemented
14#31773895C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×#16864653C2211yescharacteristic
19#22576181—24no—
20C2×#16864644C2211yescharacteristic
21C2×#16864643C2211yescharacteristic
22C2×#16864639C2211yescharacteristic
23#22576166—24no—
24C2×#16864638C2211yescharacteristic
25C2×#16864637C2211yescharacteristic
26C2×#16864637C2211yescharacteristic
27C2×C8×D271C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576140C2211yescharacteristic
30#22576139C2211yescharacteristic
31#22576138—21no—
32#22576137—21no—
33#22576136C2211yescharacteristic
34#22576135—21no—
35#22576135C2211yescharacteristic
36#22576134C2211yescharacteristic
37#22576133—21no—
38#22576133—21no—
39#22576133—21no—
40#22576132—21no—
41#22576131—21no—
42#22576131C2211yescharacteristic
43C4⋊C4×D271C2211yescharacteristic
44C4⋊C4×D271C2211yescharacteristic
45#22576092C2211yescharacteristic
46#22576091C2211yescharacteristic
47#22576091C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576083C2211yescharacteristic
51#22576082C2211yescharacteristic
52#22576081C2211yescharacteristic
53#22576080C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C22×C8×C271C2211yescharacteristic
56C22⋊Q8×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—22no—
63#16864654—22no—
64#16864654—22no—
65#16864653—22no—
66#16864653—22no—
67C2×Dic542D411yescharacteristic
68C2×Dic542C2311yescharacteristic
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084D411yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271D411yescharacteristic
77C2×C22⋊D271D411yescharacteristic
78C22×Dic271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271C2311yescharacteristic
83C2×C4×D271C2311yescharacteristic
84#16864644—22no—
85#16864643—22no—
86#16864639—22no—
87#16864638—22no—
88#16864637—22no—
89#16864637—22no—
90C2×C271⋊C8C2311yescharacteristic
91C2×C271⋊C8C2311yescharacteristic
92C8×D271—22no—
93C8×D271D414yescomplemented
94C4⋊Dic271—21no—
95C4⋊Dic271C2311yescharacteristic
96C4⋊Dic271C2311yescharacteristic
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100Dic271⋊C4—21no—
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271—21no—
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111SD16×C271—22no—
112C2×C8×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C4⋊C4×C271—21no—
116C4⋊C4×C271C2311yescharacteristic
117C4⋊C4×C271C2311yescharacteristic
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22×D271—21no—
121C22×D271C2×D411yescharacteristic
122C22×D271C2×D411yescharacteristic
123Dic542—22no—
124Dic542—21no—
125Dic542—21no—
126D1084—22no—
127D1084—21no—
128D1084—21no—
129D1084—42no—
130D1084—81no—
131C22⋊D271—22no—
132C22⋊D271—22no—
133C22⋊D271—42no—
134C22⋊D271—81no—
135C22⋊D271—81no—
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—42no—
145C4×D271—81no—
146C4×D271—81no—
147C4×D271C2×D412yes—
148C4×D271#4112yes—
149C271⋊C8—21no—
150C271⋊C8C2×D412yes—
151C23×C271—21no—
152C23×C271C2×D411yescharacteristic
153Q8×C271—21no—
154Q8×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—81no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2411yescharacteristic
165C8×C271—21no—
166C8×C271C2×D412yes—
167C22×C271—21no—
168C22×C271—21no—
169C22×C271—21no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271C22×D411yescharacteristic
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—21no—
177C2×D271—21no—
178C2×D271—21no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271#12112yes—
183C4×C271—21no—
184C4×C271—41no—
185C4×C271—41no—
186C4×C271—41no—
187C4×C271C22×D411yescharacteristic
188C4×C271C2×#4111yescharacteristic
189Dic271—21no—
190Dic271—41no—
191Dic271—41no—
192Dic271—41no—
193Dic271#12112yes—
194C2×C271—41no—
195C2×C271—81no—
196C2×C271C2×#12111yescharacteristic
197C2×C271#57711yescharacteristic
198C2×C271#57711yescharacteristic
199D271—22no—
200D271—41no—
201D271—81no—
202C271#306011yescharacteristic, complemented, Sylow
203#3060—2711nomaximal, Sylow
204C2×#135—2711no—
205C2×#131—2711no—
206#537—2711no—
207#537—2711no—
208#470—2711no—
209#470—2711no—
210#469—2711no—
211#469—2711no—
212#465—2711no—
213#465—2711no—
214#465—2711no—
215#465D27111yescharacteristic, complemented
216C2×#106—2711no—
217#417—2711no—
218#417—2711no—
219C2×#41—2711no—
220C2×#41—2711no—
221C2×#41—2711no—
222#135—5424no—
223C2×Q16—2711no—
224C2×SD16—2711no—
225C2×SD16C2×D27111yescharacteristic
226C2×D8—2711no—
227#131—5424no—
228C2×M16—2711no—
229C2×M16—2711no—
230C2×M16—2711no—
231C22×C8—2711no—
232C22×C8—2711no—
233C22×C8C2×D27111yescharacteristic
234C22⋊Q8—2711no—
235C22⋊Q8—2711no—
236C22⋊Q8—2711no—
237C22⋊Q8—5421no—
238C22⋊Q8—5421no—
239C22⋊Q8C2×D27111yescharacteristic
240#121—2711no—
241#121—2711no—
242#121—2711no—
243#121—5421no—
244#121—5421no—
245#121—5421no—
246#121—5421no—
247#121—5421no—
248#121—5421no—
249#121C2×D27111yescharacteristic
250C2×C4⋊C4—2711no—
251C2×C4⋊C4—2711no—
252#106—2711no—
253#106—2711no—
254#106—2711no—
255#106C2×D27111yescharacteristic
256#103—2711no—
257#103—2711no—
258#103—2711no—
259#103C2×D27111yescharacteristic
260#102—2711no—
261#102—2711no—
262#102—2711no—
263#102C2×D27111yescharacteristic
264#41—5422no—
265#41—5422no—
266#41—5422no—
267#41—5422no—
268#41—5422no—
269#41—5422no—
270C2×Q8—2711no—
271C2×Q8—2711no—
272C2×Q8C22×D27111yescharacteristic
273C2×D4—21no—
274C2×D4—2711no—
275C2×D4—2711no—
276C2×D4—2711no—
277C2×D4—2711no—
278C2×D4—5421no—
279C2×D4—5421no—
280C2×D4—5421no—
281C2×D4—5421no—
282C2×D4—5421no—
283C2×D4C22×D27111yescharacteristic
284C22×C4—2711no—
285C22×C4—2711no—
286C22×C4—5421no—
287C22×C4—5421no—
288C22×C4—5421no—
289C22×C4—5421no—
290C22×C4C22×D27111yescharacteristic
291Q16—5422no—
292SD16—22no—
293SD16—5422no—
294D8—5422no—
295M16—5422no—
296M16—5422no—
297M16—5422no—
298C2×C8—22no—
299C2×C8—2711no—
300C2×C8—2714no—
301C2×C8—2711no—
302C2×C8—5422no—
303C2×C8C22×D27111yescharacteristic
304C2×C8C22×D27111yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—2711no—
307C4⋊C4—2711no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4—5421no—
313C4⋊C4C22×D27111yescharacteristic
314C4⋊C4C22×D27111yescharacteristic
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C23—21no—
324C23—2711no—
325C23—2711no—
326C23—5421no—
327C23D4×D27111yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—5422no—
331Q8—5421no—
332Q8—5421no—
333D4—21no—
334D4—21no—
335D4—81no—
336D4—5422no—
337D4—5422no—
338D4—5422no—
339D4—5421no—
340D4—5421no—
341D4—10842no—
342D4—10842no—
343D4—21681no—
344D4—21681no—
345D4—21681no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—2711no—
352C2×C4—2712no—
353C2×C4—2712no—
354C2×C4—2711no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—10842no—
361C2×C4—10842no—
362C2×C4—21681no—
363C2×C4—21681no—
364C2×C4D4×D27111yescharacteristic
365C2×C4C23×D27111yescharacteristic
366C8—21no—
367C8—2712no—
368C8—5421no—
369C8D4×D27112yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—2712no—
377C22—5421no—
378C22—5421no—
379C22—5421no—
380C22—5421no—
381C22—5421no—
382C22—10841no—
383C22—10841no—
384C22—10841no—
385C22C2×D4×D27111yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—2712no—
391C4—5421no—
392C4—10841no—
393C4—10841no—
394C4—10841no—
395C4C2×D4×D27111yescharacteristic
396C4#41×D27111yescharacteristic
397C2—41no—
398C2—81no—
399C2—5422no—
400C2—10841no—
401C2—21681no—
402C2#121×D27111yescharacteristic
403C2#3177470311yescharacteristic
404C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
hypercenter#465D271
hypocenterC271#3060
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#22576181C2semidirect
2. C2×#22576166C2semidirect
3. #31774477C2semidirect
4. #31774476C2semidirect
5. #31774154C2semidirect
6. #31774153C2semidirect
7. #31774149C2semidirect
8. #31774148C2semidirect
9. #31774127C2semidirect
10. #31774126C2semidirect
11. #31774123C2semidirect
12. #106×D271C2semidirect
13. #31773897C2semidirect
14. #31773895C2semidirect
15. #465×C271C2semidirect
93. C8×D271D4semidirect
202. C271#3060semidirect
215. #465D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761812
C2×#225761662
#317744772
#317744762
#317741542
#317741532
#317741492
#317741482
#317741272
#317741262
#317741232
#106×D2712
#317738972
#317738952
#465×C2712
#3060271