Finite Group Databasebeta

← #46848101

Subgroups — #46848101

Order 34688.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576180C211yesmaximal, characteristic, complemented
2C22×#16864644C211yesmaximal, characteristic, complemented
3#31774556C211yesmaximal, characteristic, complemented
4#31774279C211yesmaximal, characteristic, complemented
5#31774136C211yesmaximal, characteristic, complemented
6#31774131C211yesmaximal, characteristic, complemented
7#31774119C211yesmaximal, characteristic, complemented
8#31774116C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774057C211yesmaximal, characteristic, complemented
11#31774036C211yesmaximal, characteristic, complemented
12#31774035C211yesmaximal, characteristic, complemented
13C4×#16864644C211yesmaximal, characteristic, complemented
14D4×C271⋊C8C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18#22576180—24no—
19C2×#16864644—24no—
20C2×#16864644C2211yescharacteristic
21C2×#16864644C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27C2×#16864641C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C22×C271⋊C8C2211yescharacteristic
30C22×C271⋊C8C2211yescharacteristic
31#22576148C2211yescharacteristic
32#22576146—21no—
33#22576139C2211yescharacteristic
34#22576138—21no—
35#22576134C2211yescharacteristic
36#22576132—21no—
37C4×Dic542C2211yescharacteristic
38(C22⋊D271)⋊C4—21no—
39#22576093C2211yescharacteristic
40C4⋊(C271⋊C8)C2211yescharacteristic
41#22576084C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576083C2211yescharacteristic
44#22576083C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576080C2211yescharacteristic
47C22⋊(C271⋊C8)C2211yescharacteristic
48C22⋊(C271⋊C8)C2211yescharacteristic
49C4×C271⋊C8C2211yescharacteristic
50C2×#41×C271C2211yescharacteristic
51C22×Q8×C271C2211yescharacteristic
52#124×C271—21no—
53#124×C271C2211yescharacteristic
54C22⋊Q8×C271—21no—
55C22⋊Q8×C271C2211yescharacteristic
56#121×C271—21no—
57#121×C271C2211yescharacteristic
58C4×Q8×C271C2211yescharacteristic
59C4×D4×C271—21no—
60C4×D4×C271C2211yescharacteristic
61#16864654—22no—
62#16864654—41no—
63C2×Dic542—22no—
64C2×Dic542—21no—
65C2×Dic542—41no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C22×Dic271—21no—
72C2×C4×D271—21no—
73#16864644—22no—
74#16864644—22no—
75#16864644—22no—
76#16864644D414yescomplemented
77#16864642—22no—
78#16864642—41no—
79#16864641—22no—
80#16864641—41no—
81#16864639—22no—
82C2×C271⋊C8—22no—
83C2×C271⋊C8—22no—
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95C4×Dic271—21no—
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271—21no—
99C2×Q8×C271—22no—
100C2×Q8×C271—41no—
101C2×Q8×C271C2311yescharacteristic
102C2×Q8×C271C2311yescharacteristic
103C2×D4×C271—21no—
104C2×D4×C271—21no—
105C2×D4×C271D411yescharacteristic
106C2×D4×C271C2311yescharacteristic
107C22×C4×C271—21no—
108C22×C4×C271—21no—
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C4⋊C4×C271—21no—
112C4⋊C4×C271—21no—
113C4⋊C4×C271D411yescharacteristic
114C4⋊C4×C271C2311yescharacteristic
115C22⋊C4×C271—21no—
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271D411yescharacteristic
120C22⋊C4×C271D411yescharacteristic
121C42×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—42no—
128Dic542C2×D412yes—
129D1084—21no—
130D1084—21no—
131C22⋊D271—42no—
132C22⋊D271—42no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—42no—
137C4×D271—42no—
138C271⋊C8—21no—
139C271⋊C8—21no—
140C271⋊C8—21no—
141C271⋊C8C2×D412yes—
142C23×C271—21no—
143C23×C271C2×D411yescharacteristic
144C23×C271C2×D411yescharacteristic
145Q8×C271—21no—
146Q8×C271—21no—
147Q8×C271—21no—
148Q8×C271—42no—
149Q8×C271C2×D412yes—
150D4×C271—22no—
151D4×C271—21no—
152D4×C271—21no—
153D4×C271—42no—
154D4×C271—42no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—42no—
163C2×C4×C271—42no—
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2×D411yescharacteristic
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2411yescharacteristic
169C22×C271—21no—
170C22×C271—21no—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271#13512yescomplemented
178C22×C271C22×D411yescharacteristic
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C4×C271—21no—
183C4×C271—21no—
184C4×C271—22no—
185C4×C271—21no—
186C4×C271—21no—
187C4×C271—41no—
188C4×C271—41no—
189C4×C271C22×D411yescharacteristic
190C4×C271H(5,𝔽2)11yescharacteristic
191Dic271—22no—
192Dic271—41no—
193Dic271—41no—
194C2×C271—22no—
195C2×C271—41no—
196C2×C271—81no—
197C2×C271D4211yescharacteristic
198C2×C271C2×#13511yescharacteristic
199C2×C271#57811yescharacteristic
200D271—81no—
201C271#328511yescharacteristic, complemented, Sylow
202#3285—2711nomaximal, Sylow
203C2×#135—2711no—
204C22×Q16—2711no—
205#548—2711no—
206#548D27111yescharacteristic, complemented
207#495—2711no—
208#467—2711no—
209#466—2711no—
210#464—2711no—
211#464—2711no—
212#452—2711no—
213#452—2711no—
214#448—2711no—
215#448—2711no—
216C4×Q16—2711no—
217C8×D4—2711no—
218C2×#41—2711no—
219C2×#41C2×D27111yescharacteristic
220C22×Q8—2711no—
221C22×Q8C2×D27111yescharacteristic
222#135—5424no—
223C2×Q16—2711no—
224C2×Q16—2711no—
225C2×Q16—2711no—
226C2×Q16—2711no—
227C2×Q16—5424no—
228C2×SD16—2711no—
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×D8—2711no—
233C22×C8—2711no—
234C22×C8—2711no—
235#124—21no—
236#124—2711no—
237#124—5421no—
238#124C2×D27111yescharacteristic
239C22⋊Q8—21no—
240C22⋊Q8—2711no—
241C22⋊Q8—5421no—
242C22⋊Q8C2×D27111yescharacteristic
243#121—21no—
244#121—2711no—
245#121—5421no—
246#121C2×D27111yescharacteristic
247C4×Q8—2711no—
248C4×Q8C2×D27111yescharacteristic
249C4×D4—21no—
250C4×D4—5421no—
251C4×D4C2×D27111yescharacteristic
252#107—2711no—
253C4⋊C8—2711no—
254#103—2711no—
255#103—2711no—
256#103—2711no—
257#103—2711no—
258#102—2711no—
259#102—2711no—
260C22⋊C8—2711no—
261C22⋊C8—2711no—
262C4×C8—2711no—
263#41—22no—
264#41—41no—
265#41—5422no—
266#41—10841no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—2711no—
271C2×Q8—2711no—
272C2×Q8—5422no—
273C2×Q8—5421no—
274C2×Q8—10841no—
275C2×Q8C22×D27111yescharacteristic
276C2×Q8C22×D27111yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—2711no—
280C2×D4—5421no—
281C2×D4—5421no—
282C2×D4C22⋊D27111yescharacteristic
283C2×D4C22×D27111yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—5421no—
287C22×C4—5421no—
288C22×C4C22×D27111yescharacteristic
289C22×C4C22×D27111yescharacteristic
290Q16—2714no—
291Q16—5422no—
292Q16—5422no—
293Q16—5422no—
294SD16—5422no—
295SD16—5422no—
296SD16—10841no—
297SD16—10841no—
298D8—5422no—
299C2×C8—2711no—
300C2×C8—2711no—
301C2×C8—2711no—
302C2×C8—2711no—
303C2×C8—5422no—
304C2×C8—5422no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—2711no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C4⋊C4C22⋊D27111yescharacteristic
311C4⋊C4C22×D27111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4C22⋊D27111yescharacteristic
321C22⋊C4C22⋊D27111yescharacteristic
322C42—21no—
323C42—5421no—
324C42C22×D27111yescharacteristic
325C23—21no—
326C23—5421no—
327C23C2×C22⋊D27111yescharacteristic
328C23C2×C22⋊D27111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—2712no—
334Q8—5421no—
335Q8—5421no—
336Q8—5421no—
337Q8—10842no—
338Q8D4×D27112yes—
339D4—22no—
340D4—21no—
341D4—21no—
342D4—42no—
343D4—42no—
344D4—5421no—
345D4—5421no—
346D4—10842no—
347D4—10842no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—10842no—
361C2×C4—10842no—
362C2×C4C2×C22⋊D27111yescharacteristic
363C2×C4C2×C22⋊D27111yescharacteristic
364C2×C4C2×C22⋊D27111yescharacteristic
365C2×C4C2×C22⋊D27111yescharacteristic
366C2×C4C23×D27111yescharacteristic
367C8—2712no—
368C8—5421no—
369C8—5421no—
370C8—5421no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—10841no—
380C22—10841no—
381C22—10841no—
382C22#2257618012yescomplemented
383C22C22×C22⋊D27111yescharacteristic
384C4—21no—
385C4—21no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—5422no—
392C4—10841no—
393C4—10841no—
394C4C2×D4×D27111yescharacteristic
395C4#2257620511yescharacteristic
396C2—22no—
397C2—41no—
398C2—81no—
399C2—21681no—
400C2D4×C22⋊D27111yescharacteristic
401C2C2×#2257618011yescharacteristic
402C2#3177471211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#548×C271C2
socleC22×C271C22×D4
hypercenter#548D271
hypocenterC271#3285
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22576180C2semidirect
2. C22×#16864644C2semidirect
3. #31774556C2semidirect
4. #31774279C2semidirect
5. #31774136C2semidirect
6. #31774131C2semidirect
7. #31774119C2semidirect
8. #31774116C2semidirect
9. #31774059C2semidirect
10. #31774057C2semidirect
11. #31774036C2semidirect
12. #31774035C2semidirect
13. C4×#16864644C2semidirect
14. D4×C271⋊C8C2semidirect
15. #548×C271C2semidirect
76. #16864644D4semidirect
177. C22×C271#135semidirect
201. C271#3285semidirect
206. #548D271semidirect
382. C22#22576180semidirect

Maximal subgroups

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

subgroupindex
C2×#225761802
C22×#168646442
#317745562
#317742792
#317741362
#317741312
#317741192
#317741162
#317740592
#317740572
#317740362
#317740352
C4×#168646442
D4×C271⋊C82
#548×C2712
#3285271