Finite Group Databasebeta

← #46839381

Subgroups — #46839381

Order 34688.

788 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576132C211yesmaximal, characteristic
2C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
3C2×(C22⋊D271)⋊C4C211yesmaximal, characteristic, complemented
4C22×Dic271⋊C4C211yesmaximal, characteristic, complemented
5#31773797C211yesmaximal, characteristic, complemented
6#31773792C211yesmaximal, characteristic, complemented
7#31773788C211yesmaximal, characteristic, complemented
8#31773785C211yesmaximal, characteristic, complemented
9#31773777C211yesmaximal, characteristic, complemented
10#31773776C211yesmaximal, characteristic, complemented
11#31773770C211yesmaximal, characteristic, complemented
12#31773768C211yesmaximal, characteristic, complemented
13C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic271C211yesmaximal, characteristic, complemented
15C2×C4×D4×C271C211yesmaximal, characteristic, complemented
16C22×C22⋊D271C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19C23×Dic271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21#22576132C418yescomplemented
22C4×C22⋊D271—24no—
23(C22⋊D271)⋊C4—24no—
24C2×Dic271⋊C4—24no—
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×C4×Dic271C2211yescharacteristic
40C2×C4×Dic271C2211yescharacteristic
41C2×C4×Dic271C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C22×D4×C271C2211yescharacteristic
49C23×C4×C271C2211yescharacteristic
50C23×C4×C271C2211yescharacteristic
51C4×D4×C271—24no—
52C2×C4⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55C2×C42×C271C2211yescharacteristic
56C23×D271C2311yescharacteristic
57C2×C22⋊D271—22no—
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271C2×C414yescomplemented
62C2×C22⋊D271C2×C414yescomplemented
63C22×Dic271—22no—
64C22×Dic271—22no—
65C22×Dic271—22no—
66C22×Dic271—22no—
67C22×Dic271—41no—
68C22×Dic271C2×C414yescomplemented
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C22×Dic271C2311yescharacteristic
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2311yescharacteristic
76C2×C4×D271—22no—
77C2×C4×D271—22no—
78C2×C4×D271—41no—
79Dic271⋊C4—22no—
80Dic271⋊C4—22no—
81Dic271⋊C4—22no—
82Dic271⋊C4—22no—
83Dic271⋊C4C2×C414yescomplemented
84Dic271⋊C4D414yescomplemented
85C22⋊Dic271—22no—
86C22⋊Dic271—22no—
87C22⋊Dic271—41no—
88C22⋊Dic271C2×C414yescomplemented
89(C2×D271)⋊C4—22no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—41no—
92(C2×D271)⋊C4C2×C414yescomplemented
93C4×Dic271—22no—
94C4×Dic271—22no—
95C4×Dic271—22no—
96C24×C271C2311yescharacteristic
97C24×C271C2311yescharacteristic
98C2×D4×C271—22no—
99C2×D4×C271—22no—
100C2×D4×C271C2×C414yescomplemented
101C22×C4×C271—22no—
102C22×C4×C271—22no—
103C22×C4×C271—22no—
104C22×C4×C271—22no—
105C22×C4×C271—41no—
106C22×C4×C271—41no—
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C4⋊C4×C271—22no—
113C22⋊C4×C271—22no—
114C22⋊C4×C271D414yescomplemented
115C42×C271—22no—
116C22×D271—21no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271C22×C412yes—
123C22⋊D271—28no—
124C22⋊D271—28no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—28no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271—44no—
142C2×Dic271C22×C412yes—
143C2×Dic271C22×C412yes—
144C2×Dic271C22×C412yes—
145C2×Dic271C2×D412yes—
146C2×Dic271C2×D412yes—
147C2×Dic271#4112yes—
148C4×D271—44no—
149C23×C271—21no—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—21no—
153C23×C271—21no—
154C23×C271—21no—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271—21no—
158C23×C271—21no—
159C23×C271C22×C412yes—
160C23×C271C22×C412yes—
161C23×C271C2×D412yes—
162C23×C271#4112yes—
163C23×C271C2411yescharacteristic
164D4×C271—28no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—44no—
177C2×C4×C271—44no—
178C2×C4×C271C22×C412yes—
179C2×C4×C271C2×D412yes—
180C2×C4×C271C2×D412yes—
181C2×C4×C271C2×D412yes—
182C22×C271—22no—
183C22×C271—22no—
184C22×C271—22no—
185C22×C271—22no—
186C22×C271—22no—
187C22×C271—22no—
188C22×C271—22no—
189C22×C271—22no—
190C22×C271—22no—
191C22×C271—41no—
192C22×C271—41no—
193C22×C271—41no—
194C22×C271—41no—
195C22×C271C4×D414yescomplemented
196C22×C271C23×C411yescharacteristic
197C22×C271C22×D411yescharacteristic
198C22×C271C22×D411yescharacteristic
199C22×C271C2×#4111yescharacteristic
200C22×C271C2×#4111yescharacteristic
201C22×C271H(5,𝔽2)11yescharacteristic
202C22×C2712−1+411yescharacteristic
203C2×D271—22no—
204C2×D271—22no—
205C2×D271—22no—
206C2×D271—41no—
207C2×D271—41no—
208C2×D271—41no—
209C2×D271—41no—
210C4×C271—22no—
211C4×C271—22no—
212C4×C271—22no—
213C4×C271—22no—
214C4×C271—41no—
215Dic271—22no—
216Dic271—22no—
217Dic271—22no—
218Dic271—41no—
219Dic271—41no—
220Dic271—41no—
221Dic271C4×D414yescomplemented
222C2×C271—24no—
223C2×C271—42no—
224C2×C271C2×C4×D411yescharacteristic
225C2×C271C2×C4×D411yescharacteristic
226C2×C271#52011yescharacteristic
227C2×C271D4211yescharacteristic
228C2×C271#54711yescharacteristic
229C2×C271#54811yescharacteristic
230C2×C271#55311yescharacteristic
231D271—42no—
232C271#236111yescharacteristic, complemented, Sylow
233#2361—2711nomaximal, Sylow
234C2×#121—2711no—
235C2×C4×D4—2711no—
236C2×C4×D4—2711no—
237C2×C4×D4D27111yescharacteristic, complemented
238C22×C4⋊C4—2711no—
239#390—2711no—
240#389—2711no—
241#388—2711no—
242#388—2711no—
243#386—2711no—
244#386—2711no—
245#385—2711no—
246#385—2711no—
247C4×C4⋊C4—2711no—
248C4×C22⋊C4—2711no—
249C22×D4—2711no—
250C22×D4—2711no—
251C22×D4C2×D27111yescharacteristic
252C23×C4—2711no—
253C23×C4—2711no—
254C23×C4—2711no—
255C23×C4C2×D27111yescharacteristic
256C23×C4C2×D27111yescharacteristic
257#121—2718no—
258C4×D4—24no—
259C4×D4—5424no—
260C4×D4—5424no—
261C2×C4⋊C4—2711no—
262C2×C4⋊C4—2711no—
263C2×C4⋊C4—2711no—
264C2×C4⋊C4—2711no—
265C2×C4⋊C4—2711no—
266C2×C4⋊C4—2711no—
267C2×C4⋊C4—5424no—
268C2×C4⋊C4C2×D27111yescharacteristic
269C2×C22⋊C4—2711no—
270C2×C22⋊C4—2711no—
271C2×C22⋊C4—2711no—
272C2×C22⋊C4—2711no—
273C2×C22⋊C4—2711no—
274C2×C22⋊C4—2711no—
275C2×C22⋊C4—2711no—
276C2×C22⋊C4—2711no—
277C2×C22⋊C4C2×D27111yescharacteristic
278C2×C22⋊C4C2×D27111yescharacteristic
279C2×C42—2711no—
280C2×C42—2711no—
281C2×C42—2711no—
282C2×C42C2×D27111yescharacteristic
283#95—2711no—
284#95—2711no—
285#95—2711no—
286#95—2711no—
287#95—2711no—
288#95—2711no—
289C24—2711no—
290C24C22×D27111yescharacteristic
291C24C22×D27111yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—2714no—
295C2×D4—2714no—
296C2×D4—5422no—
297C2×D4—5422no—
298C2×D4—5422no—
299C2×D4—5422no—
300C2×D4C4×D27114yescomplemented
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—2711no—
308C22×C4—2714no—
309C22×C4—2711no—
310C22×C4—2711no—
311C22×C4—2711no—
312C22×C4—2711no—
313C22×C4—2711no—
314C22×C4—2711no—
315C22×C4—5422no—
316C22×C4—5422no—
317C22×C4—5422no—
318C22×C4—5422no—
319C22×C4—5422no—
320C22×C4—5422no—
321C22×C4—10841no—
322C22×C4—10841no—
323C22×C4C22×D27111yescharacteristic
324C22×C4C22×D27111yescharacteristic
325C22×C4C22×D27111yescharacteristic
326C22×C4C22×D27111yescharacteristic
327C22×C4C22×D27111yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—2714no—
330C4⋊C4—2714no—
331C4⋊C4—5422no—
332C4⋊C4—5422no—
333C4⋊C4—5422no—
334C4⋊C4—5422no—
335C22⋊C4—22no—
336C22⋊C4—2714no—
337C22⋊C4—2714no—
338C22⋊C4—5422no—
339C22⋊C4—5422no—
340C22⋊C4—5422no—
341C22⋊C4—5422no—
342C22⋊C4—10841no—
343C22⋊C4—10841no—
344C22⋊C4C22⋊D27114yescomplemented
345C42—22no—
346C42—5422no—
347C42—5422no—
348C42—5422no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—2712no—
360C23—5421no—
361C23—5421no—
362C23—5421no—
363C23—5421no—
364C23—5421no—
365C23—5421no—
366C23C2×C4×D27112yes—
367C23C2×C4×D27112yes—
368C23C2×C22⋊D27112yes—
369C23#1686465412yes—
370C23C23×D27111yescharacteristic
371D4—28no—
372D4—5428no—
373D4—5428no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—2712no—
388C2×C4—2712no—
389C2×C4—2712no—
390C2×C4—2712no—
391C2×C4—2712no—
392C2×C4—2712no—
393C2×C4—5428no—
394C2×C4—5421no—
395C2×C4—5421no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—5421no—
399C2×C4—5421no—
400C2×C4—5421no—
401C2×C4—5421no—
402C2×C4—5421no—
403C2×C4—5421no—
404C2×C4—5421no—
405C2×C4—5421no—
406C2×C4—5421no—
407C2×C4—5421no—
408C2×C4—5421no—
409C2×C4—10844no—
410C2×C4—10844no—
411C2×C4C2×C4×D27112yes—
412C2×C4D4×D27112yes—
413C2×C4C2×C22⋊D27112yes—
414C2×C4C2×C22⋊D27112yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—5422no—
429C22—5422no—
430C22—5422no—
431C22—10841no—
432C22—10841no—
433C22—10841no—
434C22—10841no—
435C22C4×C22⋊D27114yescomplemented
436C22C22×C4×D27111yescharacteristic
437C22C2×D4×D27111yescharacteristic
438C22C22×C22⋊D27111yescharacteristic
439C22#41×D27111yescharacteristic
440C22C2×#1686465411yescharacteristic
441C22#2257620511yescharacteristic
442C22#2257620711yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—41no—
448C4—2714no—
449C4—5422no—
450C4—5422no—
451C4—5422no—
452C4—10841no—
453C4—10841no—
454C4—10841no—
455C2—24no—
456C2—42no—
457C2—10842no—
458C2C4×D4×D27111yescharacteristic
459C2C2×C4×C22⋊D27111yescharacteristic
460C2#3177437611yescharacteristic
461C2D4×C22⋊D27111yescharacteristic
462C2#3177454611yescharacteristic
463C2#3177455611yescharacteristic
464C2#3177459611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
FittingC2×C4×D4×C271C2
socle, cosocleC23×C271C24
prime radicalC2×#22576132C2
hypercenterC2×C4×D4D271
hypocenterC271#2361
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D271C2semidirect
3. C2×(C22⋊D271)⋊C4C2semidirect
4. C22×Dic271⋊C4C2semidirect
5. #31773797C2semidirect
6. #31773792C2semidirect
7. #31773788C2semidirect
8. #31773785C2semidirect
9. #31773777C2semidirect
10. #31773776C2semidirect
11. #31773770C2semidirect
12. #31773768C2semidirect
13. C4×Dic271⋊C4C2semidirect
14. C22⋊C4×Dic271C2semidirect
15. C2×C4×D4×C271C2semidirect
21. #22576132C4semidirect
61. C2×C22⋊D271C2×C4semidirect
62. C2×C22⋊D271C2×C4semidirect
68. C22×Dic271C2×C4semidirect
83. Dic271⋊C4C2×C4semidirect
84. Dic271⋊C4D4semidirect
88. C22⋊Dic271C2×C4semidirect
92. (C2×D271)⋊C4C2×C4semidirect
100. C2×D4×C271C2×C4semidirect
114. C22⋊C4×C271D4semidirect
195. C22×C271C4×D4semidirect
221. Dic271C4×D4semidirect
232. C271#2361semidirect
237. C2×C4×D4D271semidirect
300. C2×D4C4×D271semidirect
344. C22⋊C4C22⋊D271semidirect
435. C22C4×C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761322
C2×C4×C22⋊D2712
C2×(C22⋊D271)⋊C42
C22×Dic271⋊C42
#317737972
#317737922
#317737882
#317737852
#317737772
#317737762
#317737702
#317737682
C4×Dic271⋊C42
C22⋊C4×Dic2712
C2×C4×D4×C2712
#2361271