Finite Group Databasebeta

← #46847922

Subgroups — #46847922

Order 34688.

478 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×#22576191C211yesmaximal, characteristic, complemented
2C2×#22576183C211yesmaximal, characteristic, complemented
3#31774558C211yesmaximal, characteristic, complemented
4#31774549C211yesmaximal, characteristic, complemented
5#31774290C211yesmaximal, characteristic, complemented
6#31774145C211yesmaximal, characteristic, complemented
7#31774137C211yesmaximal, characteristic, complemented
8#31774115C211yesmaximal, characteristic, complemented
9#31774105C211yesmaximal, characteristic, complemented
10#31774062C211yesmaximal, characteristic, complemented
11#31774058C211yesmaximal, characteristic, complemented
12#31774044C211yesmaximal, characteristic, complemented
13#31774000C211yesmaximal, characteristic, complemented
14#31773971C211yesmaximal, characteristic, complemented
15#451×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864655C2211yescharacteristic
18C2×#16864654C2211yescharacteristic
19#22576191—24no—
20#22576183—24no—
21C2×#16864644C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864643C2211yescharacteristic
24C2×#16864641C2211yescharacteristic
25C2×#16864641C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27C2×D2168C2211yescharacteristic
28C2×#16864638C2211yescharacteristic
29C2×C8×D271C2211yescharacteristic
30#22576158C2211yescharacteristic
31#22576149—21no—
32#22576149C2211yescharacteristic
33#22576145—21no—
34#22576142—21no—
35#22576140C2211yescharacteristic
36#22576136—21no—
37#22576136C2211yescharacteristic
38#22576134—21no—
39#22576134C2211yescharacteristic
40#22576133—21no—
41Q8×Dic271C2211yescharacteristic
42#22576122C2211yescharacteristic
43C4×C22⋊D271—21no—
44C4×C22⋊D271C2211yescharacteristic
45C4⋊C4×D271—21no—
46#22576093C2211yescharacteristic
47Dic271⋊C8C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576082C2211yescharacteristic
51#22576080C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic
53#22576070C2211yescharacteristic
54C22×Q8×C271C2211yescharacteristic
55C2×Q16×C271C2211yescharacteristic
56C2×Q16×C271C2211yescharacteristic
57C22⋊Q8×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60C22⋊C8×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—22no—
63#16864655—41no—
64#16864655—41no—
65#16864654—22no—
66#16864654—41no—
67C2×Dic542C2311yescharacteristic
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084C2311yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271D411yescharacteristic
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271—21no—
79C2×C4×D271C2311yescharacteristic
80#16864644—22no—
81#16864644—22no—
82#16864643—41no—
83#16864641—22no—
84#16864641—22no—
85#16864641D414yescomplemented
86D2168—22no—
87#16864638—22no—
88C2×C271⋊C8C2311yescharacteristic
89C2×C271⋊C8C2311yescharacteristic
90C8×D271—22no—
91C4⋊Dic271—21no—
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4D411yescharacteristic
98C22⋊Dic271D411yescharacteristic
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4D411yescharacteristic
106C4×Dic271—21no—
107C4×Dic271C2311yescharacteristic
108C2×Q8×C271—21no—
109C2×Q8×C271—22no—
110C2×Q8×C271—41no—
111C2×Q8×C271C2311yescharacteristic
112C2×Q8×C271C2311yescharacteristic
113C2×Q8×C271C2311yescharacteristic
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271—22no—
117Q16×C271—41no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271—21no—
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271C2×D411yescharacteristic
126Dic542—21no—
127Dic542—21no—
128D1084—21no—
129D1084—21no—
130D1084—21no—
131D1084—42no—
132D1084—42no—
133D1084C2×D412yes—
134C22⋊D271—41no—
135C22⋊D271—42no—
136C22⋊D271—42no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271C2×D411yescharacteristic
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C4×D271—21no—
143C4×D271—21no—
144C4×D271—42no—
145C4×D271—42no—
146C4×D271—42no—
147C4×D271—42no—
148C271⋊C8—21no—
149C271⋊C8C2×D412yes—
150C23×C271C2×D411yescharacteristic
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271—21no—
155Q8×C271—21no—
156Q8×C271—42no—
157Q8×C271C2×D412yes—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—42no—
165C2×C4×C271C2×D411yescharacteristic
166C2×C4×C271C2411yescharacteristic
167C8×C271—21no—
168C8×C271—21no—
169C22×C271—21no—
170C22×C271—21no—
171C22×C271—21no—
172C22×C271C22×D411yescharacteristic
173C2×D271—21no—
174C2×D271—22no—
175C2×D271—21no—
176C2×D271—21no—
177C2×D271—41no—
178C2×D271—41no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C4×C271—21no—
183C4×C271—22no—
184C4×C271—41no—
185C4×C271—41no—
186C4×C271—41no—
187C4×C271C22×D411yescharacteristic
188C4×C271H(5,𝔽2)11yescharacteristic
189Dic271—21no—
190Dic271—21no—
191Dic271—21no—
192Dic271—41no—
193Dic271—41no—
194C2×C271—41no—
195C2×C271D4211yescharacteristic
196C2×C271#57511yescharacteristic
197C2×C271#57811yescharacteristic
198D271—41no—
199D271—42no—
200D271—81no—
201C271#326911yescharacteristic, complemented, Sylow
202#3269—2711nomaximal, Sylow
203C2×#137—2711no—
204C2×#135—2711no—
205#548—2711no—
206#547—2711no—
207#497—2711no—
208#468—2711no—
209#467—2711no—
210#463—2711no—
211#462—2711no—
212#452—2711no—
213#452—2711no—
214#451D27111yescharacteristic, complemented
215#449—2711no—
216#440—2711no—
217#435—2711no—
218C2×#41—2711no—
219C2×#41—2711no—
220C2×#41—2711no—
221C22×Q8C2×D27111yescharacteristic
222#137—5424no—
223#135—5424no—
224C2×Q16—2711no—
225C2×Q16—2711no—
226C2×Q16C2×D27111yescharacteristic
227C2×Q16C2×D27111yescharacteristic
228C2×SD16—2711no—
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×D8—2711no—
233C2×M16—2711no—
234C22×C8—2711no—
235C4⋊Q8—2711no—
236#124—2711no—
237#124—5421no—
238#123—5421no—
239#123—5421no—
240C22⋊Q8—2711no—
241C22⋊Q8—2711no—
242C22⋊Q8—5421no—
243C22⋊Q8C2×D27111yescharacteristic
244#121—2711no—
245#121—5421no—
246#121—5421no—
247C4×Q8—2711no—
248C4×D4—2711no—
249C4×D4—2711no—
250C4×D4—5421no—
251C2×C4⋊C4—5421no—
252#107—2711no—
253C4⋊C8—2711no—
254#103—2711no—
255#103—2711no—
256#103C2×D27111yescharacteristic
257#103C2×D27111yescharacteristic
258#102—2711no—
259#102—2711no—
260C22⋊C8—2711no—
261C22⋊C8C2×D27111yescharacteristic
262#97—2711no—
263#41—5422no—
264#41—5422no—
265#41—5422no—
266#41—10841no—
267#41—10841no—
268#41—10841no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2711no—
273C2×Q8C22×D27111yescharacteristic
274C2×Q8C22×D27111yescharacteristic
275C2×Q8C22×D27111yescharacteristic
276C2×D4—2711no—
277C2×D4—2711no—
278C2×D4—2711no—
279C2×D4—5421no—
280C2×D4—5421no—
281C2×D4—5421no—
282C2×D4—5421no—
283C22×C4—21no—
284C22×C4—2711no—
285C22×C4—5421no—
286C22×C4—5421no—
287C22×C4—5421no—
288C22×C4—5421no—
289C22×C4C22×D27111yescharacteristic
290Q16—22no—
291Q16—41no—
292Q16—5422no—
293Q16—5422no—
294SD16—2714no—
295SD16—5422no—
296SD16—5422no—
297SD16—10841no—
298D8—5422no—
299M16—5422no—
300C2×C8—2711no—
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
314C22⋊C4—21no—
315C22⋊C4—2711no—
316C22⋊C4—2711no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C42—2711no—
324C42—5421no—
325C23—2711no—
326C23—5421no—
327C23—5421no—
328C23D4×D27111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—42no—
335Q8—5421no—
336Q8—5421no—
337Q8D4×D27112yes—
338D4—2712no—
339D4—5421no—
340D4—5421no—
341D4—5421no—
342D4—10842no—
343D4—10842no—
344D4—10841no—
345D4—10842no—
346D4—10842no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—2711no—
355C2×C4—2711no—
356C2×C4—2711no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—10842no—
362C2×C4—10842no—
363C2×C4—10842no—
364C2×C4—10842no—
365C2×C4D4×D27111yescharacteristic
366C2×C4C23×D27111yescharacteristic
367C8—21no—
368C8—21no—
369C8—2712no—
370C8—5421no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—5421no—
375C22—5422no—
376C22—5421no—
377C22—5421no—
378C22—10841no—
379C22—10841no—
380C22—10841no—
381C22—10841no—
382C22—10841no—
383C22C2×D4×D27111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—5421no—
390C4—5421no—
391C4—5421no—
392C4—10841no—
393C4—10841no—
394C4C2×D4×D27111yescharacteristic
395C4#2257620511yescharacteristic
396C2—41no—
397C2—10841no—
398C2—10842no—
399C2—21681no—
400C2#3177452911yescharacteristic
401C2#3177469611yescharacteristic
402C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#451×C271C2
socleC22×C271C22×D4
hypercenter#451D271
hypocenterC271#3269
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#22576191C2semidirect
2. C2×#22576183C2semidirect
3. #31774558C2semidirect
4. #31774549C2semidirect
5. #31774290C2semidirect
6. #31774145C2semidirect
7. #31774137C2semidirect
8. #31774115C2semidirect
9. #31774105C2semidirect
10. #31774062C2semidirect
11. #31774058C2semidirect
12. #31774044C2semidirect
13. #31774000C2semidirect
14. #31773971C2semidirect
15. #451×C271C2semidirect
85. #16864641D4semidirect
201. C271#3269semidirect
214. #451D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761912
C2×#225761832
#317745582
#317745492
#317742902
#317741452
#317741372
#317741152
#317741052
#317740622
#317740582
#317740442
#317740002
#317739712
#451×C2712
#3269271