Finite Group Databasebeta

← #46847989

Subgroups — #46847989

Order 34688.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576187C211yesmaximal, characteristic, complemented
2#31774558C211yesmaximal, characteristic, complemented
3#31774535C211yesmaximal, characteristic, complemented
4#31774282C211yesmaximal, characteristic, complemented
5#31774152C211yesmaximal, characteristic, complemented
6#31774150C211yesmaximal, characteristic, complemented
7#31774121C211yesmaximal, characteristic, complemented
8#31774114C211yesmaximal, characteristic, complemented
9#31774058C211yesmaximal, characteristic, complemented
10#31774046C211yesmaximal, characteristic, complemented
11#31774042C211yesmaximal, characteristic, complemented
12#31774036C211yesmaximal, characteristic, complemented
13SD16×Dic271C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15C2×#137×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18#22576187—24no—
19C2×#16864641C2211yescharacteristic
20C2×#16864641C2211yescharacteristic
21C2×#16864641C2211yescharacteristic, complemented
22C2×D2168C2211yescharacteristic
23C2×#16864639C2211yescharacteristic
24C2×#16864637C2211yescharacteristic
25#22576149—21no—
26#22576149C2211yescharacteristic
27#22576149C2211yescharacteristic, complemented
28#22576143—21no—
29#22576136—21no—
30#22576136C2211yescharacteristic
31#22576136C2211yescharacteristic
32#22576134—21no—
33#22576134C2211yescharacteristic
34#22576134C2211yescharacteristic, complemented
35#22576132—21no—
36#22576128—21no—
37Q8×Dic271C2211yescharacteristic, complemented
38D4×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40C4×C22⋊D271C2211yescharacteristic
41C2×(C2×D271)⋊C4—21no—
42#22576092C2211yescharacteristic, complemented
43Dic271⋊C8C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576084C2211yescharacteristic, complemented
46#22576084C2211yescharacteristic, complemented
47#22576082C2211yescharacteristic
48#22576082C2211yescharacteristic, complemented
49#22576081C2211yescharacteristic
50C22⋊(C271⋊C8)C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic
52C8×Dic271C2211yescharacteristic, complemented
53C2×#41×C271C2211yescharacteristic
54C22×Q8×C271C2211yescharacteristic, complemented
55#137×C271—24no—
56C2×Q16×C271C2211yescharacteristic, complemented
57C2×Q16×C271C2211yescharacteristic, complemented
58C2×SD16×C271C2211yescharacteristic
59C2×SD16×C271C2211yescharacteristic
60C2×M16×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864655—22no—
63#16864655—41no—
64C2×D1084—21no—
65C2×D1084C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271D411yescharacteristic
71D4×D271—22no—
72D4×D271—41no—
73C22×Dic271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76#16864641—22no—
77#16864641—41no—
78#16864641—41no—
79D2168—22no—
80#16864639—22no—
81#16864637—22no—
82C2×C271⋊C8C2311yescharacteristic
83C2×C271⋊C8C2311yescharacteristic
84C4⋊Dic271—21no—
85C4⋊Dic271C2311yescharacteristic
86C4⋊Dic271C2311yescharacteristic
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91C22⋊Dic271D411yescharacteristic
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4D411yescharacteristic
101C4×Dic271—21no—
102C4×Dic271C2311yescharacteristic
103#41×C271—22no—
104#41×C271—41no—
105C2×Q8×C271—21no—
106C2×Q8×C271—22no—
107C2×Q8×C271—41no—
108C2×Q8×C271C2311yescharacteristic
109C2×Q8×C271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271—21no—
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271—22no—
117Q16×C271—22no—
118SD16×C271—22no—
119SD16×C271D414yescomplemented
120M16×C271—22no—
121C2×C8×C271C2311yescharacteristic
122C2×C8×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—81no—
128C22×D271C2×D411yescharacteristic
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084—21no—
133D1084—42no—
134C22⋊D271—41no—
135C22⋊D271—42no—
136C22⋊D271—42no—
137C22⋊D271—81no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—81no—
141C2×Dic271C2×D411yescharacteristic
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C4×D271—21no—
145C4×D271—21no—
146C4×D271—42no—
147C271⋊C8—21no—
148C271⋊C8—21no—
149C23×C271—21no—
150C23×C271C2×D411yescharacteristic
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271—21no—
155Q8×C271—21no—
156Q8×C271—42no—
157Q8×C271C2×D412yes—
158D4×C271—21no—
159D4×C271—42no—
160D4×C271C2×D412yes—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—42no—
167C2×C4×C271—42no—
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2411yescharacteristic
170C8×C271—21no—
171C8×C271C2×D412yes—
172C22×C271—21no—
173C22×C271—22no—
174C22×C271—21no—
175C22×C271—21no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271C22×D411yescharacteristic
179C2×D271—21no—
180C2×D271—21no—
181C2×D271—21no—
182C2×D271—41no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—84no—
189C4×C271—21no—
190C4×C271—22no—
191C4×C271—41no—
192C4×C271—41no—
193C4×C271C22×D411yescharacteristic
194C4×C271H(5,𝔽2)11yescharacteristic
195Dic271—21no—
196Dic271—21no—
197Dic271—21no—
198Dic271—41no—
199Dic271—41no—
200C2×C271—42no—
201C2×C271—41no—
202C2×C271D4211yescharacteristic
203C2×C271#57511yescharacteristic
204C2×C271#57711yescharacteristic
205D271—41no—
206D271—81no—
207D271—81no—
208C271#327511yescharacteristic, complemented, Sylow
209#3275—2711nomaximal, Sylow
210C2×#137D27111yescharacteristic, complemented
211C2×AGL(1,ℤ/8)—2711no—
212#548—2711no—
213#546—2711no—
214#495—2711no—
215#470—2711no—
216#469—2711no—
217#464—2711no—
218#463—2711no—
219#452—2711no—
220#450—2711no—
221#449—2711no—
222#448—2711no—
223C4×SD16—2711no—
224#436—2711no—
225C2×#41—2711no—
226C2×#41C2×D27111yescharacteristic
227C22×Q8C2×D27111yescharacteristic, complemented
228C22×D4—2711no—
229#137—24no—
230AGL(1,ℤ/8)—5424no—
231C2×Q16C2×D27111yescharacteristic, complemented
232C2×Q16C2×D27111yescharacteristic, complemented
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16C2×D27111yescharacteristic
237C2×SD16C2×D27111yescharacteristic
238C2×D8—2711no—
239C2×D8—2711no—
240C2×M16—2711no—
241C2×M16C2×D27111yescharacteristic, complemented
242#124—2711no—
243#124—2711no—
244#124—5421no—
245#123—5421no—
246C22⋊Q8—2711no—
247C22⋊Q8—2711no—
248C22⋊Q8—5421no—
249#121—2711no—
250#121—2711no—
251#121—5421no—
252#121—5421no—
253#120—5421no—
254C4×Q8—2711no—
255C4×D4—2711no—
256C4×D4—2711no—
257C4×D4—5421no—
258C2×C22⋊C4—5421no—
259#106—2711no—
260C4⋊C8—2711no—
261#103—2711no—
262#103—2711no—
263#103—2711no—
264#102—2711no—
265#102—2711no—
266#102—2711no—
267C22⋊C8—2711no—
268C22⋊C8—2711no—
269C4×C8—2711no—
270C24—5421no—
271#41—22no—
272#41—41no—
273#41—5422no—
274#41—10841no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D27111yescharacteristic
279C2×Q8C22×D27111yescharacteristic
280C2×Q8C22×D27111yescharacteristic
281C2×D4—21no—
282C2×D4—2711no—
283C2×D4—2711no—
284C2×D4—2711no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4—5422no—
288C2×D4—5421no—
289C2×D4—5421no—
290C2×D4—10841no—
291C2×D4C22×D27111yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—2711no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4C22×D27111yescharacteristic
298Q16—22no—
299Q16—22no—
300SD16—22no—
301SD16—5422no—
302SD16—10841no—
303SD16—10841no—
304SD16C22⋊D27114yescomplemented
305D8—5422no—
306D8—5422no—
307M16—22no—
308M16—5422no—
309C2×C8—2711no—
310C2×C8—2711no—
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—2711no—
314C4⋊C4—2711no—
315C4⋊C4—2711no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C4⋊C4—5421no—
319C22⋊C4—2711no—
320C22⋊C4—2711no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4—5421no—
330C42—2711no—
331C42—5421no—
332C23—21no—
333C23—2711no—
334C23—5421no—
335C23—5421no—
336C23—10842no—
337C23—10842no—
338C23—21681no—
339C23D4×D27111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—42no—
346Q8C2×C22⋊D27112yes—
347D4—21no—
348D4—42no—
349D4—5421no—
350D4—5421no—
351D4—5421no—
352D4—5421no—
353D4—10842no—
354D4—10841no—
355D4—10842no—
356D4—10842no—
357D4—21681no—
358D4C2×C22⋊D27112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—2711no—
367C2×C4—2711no—
368C2×C4—2711no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—5421no—
373C2×C4—10842no—
374C2×C4—21681no—
375C2×C4D4×D27111yescharacteristic
376C2×C4C23×D27111yescharacteristic
377C8—21no—
378C8—5421no—
379C8—5421no—
380C8C2×C22⋊D27112yes—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—5421no—
388C22—5421no—
389C22—5421no—
390C22—10841no—
391C22—10841no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22—10841no—
396C22—21684no—
397C22C2×D4×D27111yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—5421no—
403C4—5421no—
404C4—5421no—
405C4—10841no—
406C4—10841no—
407C4C22×C22⋊D27111yescharacteristic
408C4#2257620511yescharacteristic
409C2—41no—
410C2—42no—
411C2—10841no—
412C2—21681no—
413C2—21681no—
414C2D4×C22⋊D27111yescharacteristic
415C2#3177469611yescharacteristic
416C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
hypercenterC2×#137D271
hypocenterC271#3275
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. #31774558C2semidirect
3. #31774535C2semidirect
4. #31774282C2semidirect
5. #31774152C2semidirect
6. #31774150C2semidirect
7. #31774121C2semidirect
8. #31774114C2semidirect
9. #31774058C2semidirect
10. #31774046C2semidirect
11. #31774042C2semidirect
12. #31774036C2semidirect
13. SD16×Dic271C2semidirect
14. #31773973C2semidirect
15. C2×#137×C271C2semidirect
21. C2×#16864641C22semidirect
27. #22576149C22semidirect
34. #22576134C22semidirect
37. Q8×Dic271C22semidirect
42. #22576092C22semidirect
43. Dic271⋊C8C22semidirect
44. #22576084C22semidirect
45. #22576084C22semidirect
46. #22576084C22semidirect
48. #22576082C22semidirect
50. C22⋊(C271⋊C8)C22semidirect
52. C8×Dic271C22semidirect
54. C22×Q8×C271C22semidirect
56. C2×Q16×C271C22semidirect
57. C2×Q16×C271C22semidirect
60. C2×M16×C271C22semidirect
119. SD16×C271D4semidirect
208. C271#3275semidirect
210. C2×#137D271semidirect
227. C22×Q8C2×D271semidirect
231. C2×Q16C2×D271semidirect
232. C2×Q16C2×D271semidirect
241. C2×M16C2×D271semidirect
304. SD16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
#317745582
#317745352
#317742822
#317741522
#317741502
#317741212
#317741142
#317740582
#317740462
#317740422
#317740362
SD16×Dic2712
#317739732
C2×#137×C2712
#3275271