Finite Group Databasebeta

← #46846048

Subgroups — #46846048

Order 34688.

472 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576189C211yesmaximal, characteristic, complemented
2C2×#22576166C211yesmaximal, characteristic, complemented
3#31774476C211yesmaximal, characteristic
4#31774470C211yesmaximal, characteristic, complemented
5#31774156C211yesmaximal, characteristic, complemented
6#31774153C211yesmaximal, characteristic, complemented
7#31774141C211yesmaximal, characteristic, complemented
8#31774140C211yesmaximal, characteristic, complemented
9#31774138C211yesmaximal, characteristic, complemented
10#31774137C211yesmaximal, characteristic, complemented
11#31774125C211yesmaximal, characteristic, complemented
12#31773936C211yesmaximal, characteristic, complemented
13#31773890C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576189—24no—
20C2×Dic1084C2211yescharacteristic
21C2×#16864644C2211yescharacteristic
22C2×#16864642C2211yescharacteristic
23#22576166—24no—
24C2×#16864638C2211yescharacteristic
25C2×#16864637C2211yescharacteristic
26C2×#16864637C2211yescharacteristic
27C2×C8×D271C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576159—21no—
30#22576146—21no—
31#22576145—21no—
32#22576142—21no—
33#22576140C2211yescharacteristic
34#22576140C2211yescharacteristic
35#22576139C2211yescharacteristic
36#22576139C2211yescharacteristic
37#22576137—21no—
38#22576136C2211yescharacteristic
39#22576133—21no—
40#22576132—21no—
41#22576131—21no—
42#22576131C2211yescharacteristic
43#22576118C2211yescharacteristic
44C4⋊C4×D271C2211yescharacteristic
45#22576094C2211yescharacteristic
46#22576093C2211yescharacteristic
47#22576091C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576085C2211yescharacteristic
50#22576084C2211yescharacteristic
51#22576083C2211yescharacteristic
52#22576083C2211yescharacteristic
53#22576081C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C22×C8×C271C2211yescharacteristic
56C22⋊Q8×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic
61#16864654—22no—
62#16864654—22no—
63#16864653—22no—
64#16864653—22no—
65C2×Dic542—21no—
66C2×Dic542D411yescharacteristic
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69Q8×D271—22no—
70Q8×D271—22no—
71C2×D1084D411yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271D411yescharacteristic
75C2×C22⋊D271D411yescharacteristic
76C22×Dic271—21no—
77C2×C4×D271—21no—
78C2×C4×D271—21no—
79C2×C4×D271C2311yescharacteristic
80C2×C4×D271C2311yescharacteristic
81Dic1084—22no—
82#16864644—22no—
83#16864642—22no—
84#16864638—22no—
85#16864637—22no—
86#16864637D414yescomplemented
87C2×C271⋊C8C2311yescharacteristic
88C2×C271⋊C8C2311yescharacteristic
89C8×D271—22no—
90C8×D271—22no—
91C4⋊Dic271—21no—
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94C4⋊Dic271C2311yescharacteristic
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106C4×Dic271—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×D271C2×D411yescharacteristic
121C22×D271C2×D411yescharacteristic
122Dic542—22no—
123Dic542—21no—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—42no—
128D1084—22no—
129C22⋊D271—22no—
130C22⋊D271—22no—
131C22⋊D271—42no—
132C22⋊D271—81no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—42no—
138C2×Dic271C2×D411yescharacteristic
139C2×Dic271C2×D411yescharacteristic
140C4×D271—21no—
141C4×D271—21no—
142C4×D271—42no—
143C4×D271—81no—
144C4×D271C2×D412yes—
145C4×D271#4112yes—
146C271⋊C8—21no—
147C271⋊C8C2×D412yes—
148C23×C271—21no—
149C23×C271C2×D411yescharacteristic
150Q8×C271—21no—
151Q8×C271—21no—
152D4×C271—21no—
153D4×C271—21no—
154D4×C271—81no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2411yescharacteristic
162C8×C271—21no—
163C8×C271C2×D412yes—
164C22×C271—21no—
165C22×C271—21no—
166C22×C271—21no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271C22×D411yescharacteristic
171C2×D271—21no—
172C2×D271—21no—
173C2×D271—21no—
174C2×D271—21no—
175C2×D271—21no—
176C2×D271#12112yes—
177C4×C271—21no—
178C4×C271—41no—
179C4×C271—41no—
180C4×C271—41no—
181C4×C271C22×D411yescharacteristic
182C4×C271C2×#4111yescharacteristic
183Dic271—21no—
184Dic271—41no—
185Dic271—41no—
186Dic271—41no—
187Dic271—41no—
188Dic271#12112yes—
189C2×C271—41no—
190C2×C271—81no—
191C2×C271C2×#12111yescharacteristic
192C2×C271#57711yescharacteristic
193C2×C271#57811yescharacteristic
194D271—22no—
195D271—41no—
196C271#306311yescharacteristic, complemented, Sylow
197#3063—2711nomaximal, Sylow
198C2×#137—2711no—
199C2×#131—2711no—
200#537—2711no—
201#536—2711no—
202#470—2711no—
203#470—2711no—
204#468—2711no—
205#468—2711no—
206#467—2711no—
207#467—2711no—
208#465—2711no—
209#465D27111yescharacteristic, complemented
210#428—2711no—
211#416—2711no—
212C2×#103—2711no—
213C2×#41—2711no—
214C2×#41—2711no—
215C22×Q8—2711no—
216#137—5424no—
217C2×Q16—2711no—
218C2×Q16—2711no—
219C2×SD16—2711no—
220C2×SD16C2×D27111yescharacteristic
221#131—5424no—
222C2×M16—2711no—
223C2×M16—2711no—
224C2×M16—2711no—
225C22×C8—2711no—
226C22×C8—2711no—
227C22×C8C2×D27111yescharacteristic
228C4⋊Q8—5421no—
229#124—5421no—
230#123—5421no—
231#123—5421no—
232C22⋊Q8—2711no—
233C22⋊Q8—2711no—
234C22⋊Q8—2711no—
235C22⋊Q8—2711no—
236C22⋊Q8—2711no—
237C22⋊Q8—5421no—
238C22⋊Q8C2×D27111yescharacteristic
239#121—2711no—
240#121—5421no—
241#121—5421no—
242#121—5421no—
243#121C2×D27111yescharacteristic
244#117—2711no—
245C2×C4⋊C4—2711no—
246#107—2711no—
247#107—2711no—
248#106—2711no—
249#106C2×D27111yescharacteristic
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103—2711no—
255#103C2×D27111yescharacteristic
256#102—2711no—
257#102C2×D27111yescharacteristic
258#41—5422no—
259#41—5422no—
260#41—5422no—
261#41—5422no—
262C2×Q8—2711no—
263C2×Q8—2711no—
264C2×Q8—2711no—
265C2×Q8—5421no—
266C2×Q8—5422no—
267C2×Q8—5422no—
268C2×Q8C22×D27111yescharacteristic
269C2×D4—21no—
270C2×D4—2711no—
271C2×D4—2711no—
272C2×D4—2711no—
273C2×D4—5421no—
274C2×D4—5421no—
275C2×D4C22×D27111yescharacteristic
276C22×C4—2711no—
277C22×C4—2711no—
278C22×C4—5421no—
279C22×C4—5421no—
280C22×C4—5421no—
281C22×C4C22×D27111yescharacteristic
282Q16—5422no—
283Q16—5422no—
284SD16—22no—
285SD16—5422no—
286M16—2714no—
287M16—5422no—
288M16—5422no—
289C2×C8—22no—
290C2×C8—2711no—
291C2×C8—2711no—
292C2×C8—5422no—
293C2×C8—5422no—
294C2×C8C22×D27111yescharacteristic
295C2×C8C22×D27111yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—2711no—
298C4⋊C4—2711no—
299C4⋊C4—5421no—
300C4⋊C4—5421no—
301C4⋊C4—5421no—
302C4⋊C4—5421no—
303C4⋊C4—5421no—
304C4⋊C4—5421no—
305C4⋊C4C22×D27111yescharacteristic
306C4⋊C4C22×D27111yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—5421no—
310C22⋊C4—5421no—
311C22⋊C4—5421no—
312C22⋊C4—5421no—
313C22⋊C4—5421no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C42—5421no—
317C23—21no—
318C23—2711no—
319C23—2711no—
320C23D4×D27111yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—5422no—
324Q8—5421no—
325Q8—5421no—
326Q8—5421no—
327Q8—5421no—
328Q8—10842no—
329D4—21no—
330D4—21no—
331D4—81no—
332D4—5422no—
333D4—5422no—
334D4—5422no—
335D4—10842no—
336D4—21681no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2711no—
343C2×C4—2712no—
344C2×C4—2712no—
345C2×C4—2711no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—10842no—
353C2×C4—10842no—
354C2×C4—21681no—
355C2×C4D4×D27111yescharacteristic
356C2×C4C23×D27111yescharacteristic
357C8—21no—
358C8—2712no—
359C8—5421no—
360C8D4×D27112yes—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—2712no—
368C22—5421no—
369C22—5421no—
370C22—5421no—
371C22—5421no—
372C22—5421no—
373C22C2×D4×D27111yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—2712no—
379C4—5421no—
380C4—10841no—
381C4—10841no—
382C4—10841no—
383C4—10841no—
384C4C2×D4×D27111yescharacteristic
385C4#41×D27111yescharacteristic
386C2—41no—
387C2—81no—
388C2—5422no—
389C2—10841no—
390C2#121×D27111yescharacteristic
391C2#3177470311yescharacteristic
392C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
prime radical#31774476C2
hypercenter#465D271
hypocenterC271#3063
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×#22576189C2semidirect
2. C2×#22576166C2semidirect
4. #31774470C2semidirect
5. #31774156C2semidirect
6. #31774153C2semidirect
7. #31774141C2semidirect
8. #31774140C2semidirect
9. #31774138C2semidirect
10. #31774137C2semidirect
11. #31774125C2semidirect
12. #31773936C2semidirect
13. #31773890C2semidirect
14. #103×D271C2semidirect
15. #465×C271C2semidirect
86. #16864637D4semidirect
196. C271#3063semidirect
209. #465D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
C2×#225761662
#317744762
#317744702
#317741562
#317741532
#317741412
#317741402
#317741382
#317741372
#317741252
#317739362
#317738902
#103×D2712
#465×C2712
#3063271