Finite Group Databasebeta

← #46839823

Subgroups — #46839823

Order 34688.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576128C211yesmaximal, characteristic
2C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D271C211yesmaximal, characteristic, complemented
4#31773832C211yesmaximal, characteristic, complemented
5#31773825C211yesmaximal, characteristic, complemented
6#31773813C211yesmaximal, characteristic, complemented
7#31773811C211yesmaximal, characteristic, complemented
8#31773790C211yesmaximal, characteristic, complemented
9#31773787C211yesmaximal, characteristic, complemented
10#31773776C211yesmaximal, characteristic, complemented
11#31773773C211yesmaximal, characteristic, complemented
12#31773768C211yesmaximal, characteristic, complemented
13#31773754C211yesmaximal, characteristic, complemented
14#31773752C211yesmaximal, characteristic, complemented
15C2×#123×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21#22576128—24no—
22C4×C22⋊D271—24no—
23C2×C4⋊Dic271C2211yescharacteristic, complemented
24C2×Dic271⋊C4C2211yescharacteristic, complemented
25C2×Dic271⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×C22⋊Dic271C2211yescharacteristic, complemented
30C2×C22⋊Dic271C2211yescharacteristic, complemented
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C22⋊C4×D271—24no—
40C2×C4×Dic271C2211yescharacteristic, complemented
41C271⋊#95C2211yescharacteristic, complemented
42C271⋊#95C2211yescharacteristic, complemented
43C271⋊#95C2211yescharacteristic, complemented
44C271⋊#95C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C22×D4×C271C2211yescharacteristic
48C23×C4×C271C2211yescharacteristic, complemented
49#123×C271—24no—
50C2×C4⋊C4×C271C2211yescharacteristic, complemented
51C2×C4⋊C4×C271C2211yescharacteristic, complemented
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic, complemented
55C23×D271—22no—
56C23×D271—22no—
57C23×D271—22no—
58C23×D271—22no—
59C23×D271—22no—
60C23×D271—41no—
61C23×D271—41no—
62C23×D271C2311yescharacteristic
63C23×D271C2311yescharacteristic
64C2×C22⋊D271—22no—
65C2×C22⋊D271—22no—
66C2×C22⋊D271—22no—
67C2×C22⋊D271—22no—
68C2×C22⋊D271—41no—
69C2×C22⋊D271D414yes—
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C22×Dic271C2311yescharacteristic
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2311yescharacteristic
76C2×C4×D271—22no—
77C2×C4×D271—22no—
78C2×C4×D271—22no—
79C2×C4×D271—22no—
80C2×C4×D271—41no—
81C2×C4×D271—41no—
82C4⋊Dic271—41no—
83Dic271⋊C4—22no—
84Dic271⋊C4—41no—
85C22⋊Dic271—22no—
86C22⋊Dic271—22no—
87C22⋊Dic271—22no—
88C22⋊Dic271—41no—
89C22⋊Dic271—41no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—22no—
94(C2×D271)⋊C4—22no—
95(C2×D271)⋊C4—22no—
96(C2×D271)⋊C4—22no—
97(C2×D271)⋊C4—41no—
98C4×Dic271—22no—
99C24×C271C2311yescharacteristic
100C24×C271C2311yescharacteristic
101C2×D4×C271—22no—
102C2×D4×C271—22no—
103C2×D4×C271—41no—
104C22×C4×C271—22no—
105C22×C4×C271—22no—
106C22×C4×C271—41no—
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C4⋊C4×C271—22no—
113C4⋊C4×C271—22no—
114C22⋊C4×C271—22no—
115C22⋊C4×C271—22no—
116C22⋊C4×C271D414yescomplemented
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—28no—
122C22×D271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—44no—
129C22×D271—44no—
130C22×D271—44no—
131C22×D271—44no—
132C22×D271—44no—
133C22×D271—44no—
134C22×D271C2×D412yes—
135C22×D271#4112yes—
136C22×D271#4112yes—
137C22×D271#4112yes—
138C22⋊D271—28no—
139C22⋊D271—44no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—21no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271C2×D412yes—
158C4×D271—44no—
159C4×D271—44no—
160C23×C271—21no—
161C23×C271—21no—
162C23×C271—21no—
163C23×C271—21no—
164C23×C271—21no—
165C23×C271—21no—
166C23×C271—21no—
167C23×C271—21no—
168C23×C271—21no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271—21no—
172C23×C271C2×D412yes—
173C23×C271C2×D412yes—
174C23×C271C2411yescharacteristic
175D4×C271—44no—
176C2×C4×C271—21no—
177C2×C4×C271—21no—
178C2×C4×C271—21no—
179C2×C4×C271—21no—
180C2×C4×C271—21no—
181C2×C4×C271—21no—
182C2×C4×C271—21no—
183C2×C4×C271—21no—
184C2×C4×C271—21no—
185C2×C4×C271—21no—
186C2×C4×C271—21no—
187C2×C4×C271—21no—
188C2×C4×C271—21no—
189C2×C4×C271—44no—
190C2×C4×C271C2×D412yes—
191C2×C4×C271C2×D412yes—
192C22×C271—22no—
193C22×C271—22no—
194C22×C271—22no—
195C22×C271—22no—
196C22×C271—22no—
197C22×C271—22no—
198C22×C271—41no—
199C22×C271—41no—
200C22×C271—41no—
201C22×C271—41no—
202C22×C271—41no—
203C22×C271—41no—
204C22×C271—41no—
205C22×C271—41no—
206C22×C271C22×D411yescharacteristic
207C22×C271C22×D411yescharacteristic
208C22×C271C2×#4111yescharacteristic
209C22×C271C2×#4111yescharacteristic
210C22×C271C2×#4111yescharacteristic
211C22×C271H(5,𝔽2)11yescharacteristic
212C22×C271H(5,𝔽2)11yescharacteristic
213C2×D271—22no—
214C2×D271—22no—
215C2×D271—22no—
216C2×D271—22no—
217C2×D271—22no—
218C2×D271—22no—
219C2×D271—22no—
220C2×D271—22no—
221C2×D271—22no—
222C2×D271—416no—
223C2×D271—41no—
224C2×D271—41no—
225C2×D271—41no—
226C2×D271—41no—
227C2×D271#12314yescomplemented
228C4×C271—22no—
229C4×C271—22no—
230C4×C271—41no—
231C4×C271—41no—
232C4×C271—41no—
233Dic271—22no—
234Dic271—41no—
235Dic271—41no—
236Dic271—41no—
237Dic271—41no—
238Dic271—41no—
239C2×C271—42no—
240C2×C271—42no—
241C2×C271C2×#12311yescharacteristic
242C2×C271#52511yescharacteristic
243C2×C271#53811yescharacteristic
244C2×C271D4211yescharacteristic
245C2×C271#54611yescharacteristic
246C2×C271#54611yescharacteristic
247C2×C271#55111yescharacteristic
248D271—24no—
249D271—42no—
250C271#241811yescharacteristic, complemented, Sylow
251#2418—2711nomaximal, Sylow
252C2×#123D27111yescharacteristic, complemented
253C2×#120—2711no—
254C2×C4×D4—2711no—
255C22×C22⋊C4—2711no—
256#399—2711no—
257#397—2711no—
258#394—2711no—
259#394—2711no—
260#388—2711no—
261#388—2711no—
262#386—2711no—
263#386—2711no—
264#385—2711no—
265#380—2711no—
266#379—2711no—
267C25—2711no—
268C22×D4—2711no—
269C22×D4—2711no—
270C22×D4C2×D27111yescharacteristic
271C23×C4—2711no—
272C23×C4—2711no—
273C23×C4C2×D27111yescharacteristic, complemented
274#123—24no—
275#120—5424no—
276C4×D4—5424no—
277C2×C4⋊C4—2711no—
278C2×C4⋊C4—2711no—
279C2×C4⋊C4—2711no—
280C2×C4⋊C4C2×D27111yescharacteristic, complemented
281C2×C4⋊C4C2×D27111yescharacteristic, complemented
282C2×C22⋊C4—2711no—
283C2×C22⋊C4—2711no—
284C2×C22⋊C4—2711no—
285C2×C22⋊C4—2711no—
286C2×C22⋊C4—2711no—
287C2×C22⋊C4—2711no—
288C2×C22⋊C4—2711no—
289C2×C22⋊C4—2711no—
290C2×C22⋊C4—2711no—
291C2×C22⋊C4—2711no—
292C2×C22⋊C4—2711no—
293C2×C22⋊C4—2711no—
294C2×C22⋊C4—2711no—
295C2×C22⋊C4—5424no—
296C2×C22⋊C4C2×D27111yescharacteristic
297C2×C22⋊C4C2×D27111yescharacteristic
298C2×C22⋊C4C2×D27111yescharacteristic, complemented
299C2×C42—2711no—
300#95—2711no—
301#95—2711no—
302#95—2711no—
303#95—2711no—
304#95—2711no—
305#95—2711no—
306C24—2711no—
307C24—2711no—
308C24—5422no—
309C24—5422no—
310C24—5422no—
311C24—5422no—
312C24—5422no—
313C24—10841no—
314C24—10841no—
315C24C22×D27111yescharacteristic
316C24C22×D27111yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—2714no—
321C2×D4—5422no—
322C2×D4—5422no—
323C2×D4—5422no—
324C2×D4—5422no—
325C2×D4—10841no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—2711no—
330C22×C4—2711no—
331C22×C4—2711no—
332C22×C4—2711no—
333C22×C4—2711no—
334C22×C4—2711no—
335C22×C4—5422no—
336C22×C4—5422no—
337C22×C4—5422no—
338C22×C4—5422no—
339C22×C4—10841no—
340C22×C4—10841no—
341C22×C4C22×D27111yescharacteristic
342C22×C4C22×D27111yescharacteristic
343C22×C4C22×D27111yescharacteristic
344C22×C4C22×D27111yescharacteristic
345C22×C4C22×D27111yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—5422no—
349C4⋊C4—10841no—
350C4⋊C4—10841no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—5422no—
354C22⋊C4—5422no—
355C22⋊C4—5422no—
356C22⋊C4—5422no—
357C22⋊C4—5422no—
358C22⋊C4—5422no—
359C22⋊C4—5422no—
360C22⋊C4—5422no—
361C22⋊C4—5422no—
362C22⋊C4—5422no—
363C22⋊C4—10841no—
364C22⋊C4—10841no—
365C22⋊C4—10841no—
366C22⋊C4C22⋊D27114yescomplemented
367C42—5422no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—2712no—
381C23—2712no—
382C23—2712no—
383C23—2712no—
384C23—5428no—
385C23—5421no—
386C23—5421no—
387C23—5421no—
388C23—5421no—
389C23—5421no—
390C23—5421no—
391C23—5421no—
392C23—5421no—
393C23—5421no—
394C23—5421no—
395C23—10844no—
396C23—10844no—
397C23—10844no—
398C23—10844no—
399C23—10844no—
400C23—10844no—
401C23D4×D27112yes—
402C23C2×C22⋊D27112yes—
403C23C23×D27111yescharacteristic
404D4—44no—
405D4—5428no—
406D4—10844no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—44no—
421C2×C4—2712no—
422C2×C4—5421no—
423C2×C4—5421no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4—5421no—
429C2×C4—5421no—
430C2×C4—5421no—
431C2×C4—5421no—
432C2×C4—5421no—
433C2×C4—5421no—
434C2×C4—5421no—
435C2×C4—5421no—
436C2×C4—5421no—
437C2×C4—5421no—
438C2×C4—5421no—
439C2×C4—10844no—
440C2×C4—10844no—
441C2×C4C2×C22⋊D27112yes—
442C2×C4C2×C22⋊D27112yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—2714no—
458C22—5422no—
459C22—5422no—
460C22—5422no—
461C22—5422no—
462C22—5422no—
463C22—5422no—
464C22—5422no—
465C22—5422no—
466C22—5422no—
467C22—108416no—
468C22—10841no—
469C22—10841no—
470C22—10841no—
471C22—10841no—
472C22C2×D4×D27111yescharacteristic
473C22C22×C22⋊D27111yescharacteristic
474C22#41×D27111yescharacteristic
475C22#41×D27111yescharacteristic
476C22#41×D27111yescharacteristic
477C22#2257620511yescharacteristic
478C22#2257620611yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—5422no—
485C4—10841no—
486C4—10841no—
487C4—10841no—
488C4—10841no—
489C4—10841no—
490C2—42no—
491C2—42no—
492C2—5424no—
493C2—10842no—
494C2#123×D27111yescharacteristic
495C2#3177440411yescharacteristic
496C2#3177448311yescharacteristic
497C2D4×C22⋊D27111yescharacteristic
498C2#3177453511yescharacteristic
499C2#3177453911yescharacteristic
500C2#3177457511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#123×C271C2
prime radicalC2×#22576128C2
hypercenterC2×#123D271
hypocenterC271#2418
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D271C2semidirect
3. C2×C22⋊C4×D271C2semidirect
4. #31773832C2semidirect
5. #31773825C2semidirect
6. #31773813C2semidirect
7. #31773811C2semidirect
8. #31773790C2semidirect
9. #31773787C2semidirect
10. #31773776C2semidirect
11. #31773773C2semidirect
12. #31773768C2semidirect
13. #31773754C2semidirect
14. #31773752C2semidirect
15. C2×#123×C271C2semidirect
23. C2×C4⋊Dic271C22semidirect
24. C2×Dic271⋊C4C22semidirect
25. C2×Dic271⋊C4C22semidirect
29. C2×C22⋊Dic271C22semidirect
30. C2×C22⋊Dic271C22semidirect
40. C2×C4×Dic271C22semidirect
41. C271⋊#95C22semidirect
42. C271⋊#95C22semidirect
43. C271⋊#95C22semidirect
44. C271⋊#95C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
48. C23×C4×C271C22semidirect
50. C2×C4⋊C4×C271C22semidirect
51. C2×C4⋊C4×C271C22semidirect
54. C2×C22⋊C4×C271C22semidirect
116. C22⋊C4×C271D4semidirect
227. C2×D271#123semidirect
250. C271#2418semidirect
252. C2×#123D271semidirect
273. C23×C4C2×D271semidirect
280. C2×C4⋊C4C2×D271semidirect
281. C2×C4⋊C4C2×D271semidirect
298. C2×C22⋊C4C2×D271semidirect
366. C22⋊C4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761282
C2×C4×C22⋊D2712
C2×C22⋊C4×D2712
#317738322
#317738252
#317738132
#317738112
#317737902
#317737872
#317737762
#317737732
#317737682
#317737542
#317737522
C2×#123×C2712
#2418271