Finite Group Databasebeta

← #46834608

Subgroups — #46834608

Order 34688.

780 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic271⋊C4C211yesmaximal, characteristic, complemented
2C22×(C2×D271)⋊C4C211yesmaximal, characteristic
3C22×(C2×D271)⋊C4C211yesmaximal, characteristic
4C22×(C2×D271)⋊C4C211yesmaximal, characteristic
5C2×C271⋊#95C211yesmaximal, characteristic, complemented
6C2×C271⋊#95C211yesmaximal, characteristic, complemented
7C2×#95×C271C211yesmaximal, characteristic, complemented
8C24×D271C2211yescharacteristic
9C23×Dic271C2211yescharacteristic
10C23×Dic271C2211yescharacteristic
11C23×Dic271C2211yescharacteristic
12C2×Dic271⋊C4—22no—
13C2×Dic271⋊C4—22no—
14C2×Dic271⋊C4—22no—
15C2×(C2×D271)⋊C4—22no—
16C2×(C2×D271)⋊C4—22no—
17C2×(C2×D271)⋊C4—22no—
18C2×(C2×D271)⋊C4—22no—
19C2×(C2×D271)⋊C4—22no—
20C2×(C2×D271)⋊C4—22no—
21C2×(C2×D271)⋊C4—22no—
22C2×(C2×D271)⋊C4—22no—
23C2×(C2×D271)⋊C4C414yescomplemented
24C271⋊#95—22no—
25C271⋊#95—22no—
26C23×C4×C271C2211yescharacteristic
27C23×C4×C271C2211yescharacteristic
28C23×C4×C271C2211yescharacteristic
29#95×C271—22no—
30C23×D271—21no—
31C23×D271—21no—
32C23×D271—21no—
33C23×D271—21no—
34C23×D271—21no—
35C23×D271—21no—
36C23×D271—21no—
37C23×D271—21no—
38C23×D271—21no—
39C23×D271—21no—
40C23×D271—21no—
41C23×D271—21no—
42C23×D271C2×C412yes—
43C23×D271D412yes—
44C23×D271Q812yes—
45C22×Dic271—21no—
46C22×Dic271—21no—
47C22×Dic271—21no—
48C22×Dic271—21no—
49C22×Dic271—21no—
50C22×Dic271—21no—
51C22×Dic271—21no—
52C22×Dic271—21no—
53C22×Dic271—21no—
54C22×Dic271—21no—
55C22×Dic271—21no—
56C22×Dic271—21no—
57C22×Dic271—21no—
58C22×Dic271—21no—
59C22×Dic271—21no—
60C22×Dic271—21no—
61C22×Dic271C2×C412yes—
62C22×Dic271D412yes—
63C22×Dic271D412yes—
64C22×Dic271D412yes—
65C22×Dic271D412yes—
66Dic271⋊C4—28no—
67(C2×D271)⋊C4—28no—
68(C2×D271)⋊C4—28no—
69(C2×D271)⋊C4—28no—
70C24×C271C2311yescharacteristic
71C22×C4×C271—21no—
72C22×C4×C271—21no—
73C22×C4×C271—21no—
74C22×C4×C271—21no—
75C22×C4×C271—21no—
76C22×C4×C271—21no—
77C22×C4×C271—21no—
78C22×C4×C271—21no—
79C22×C4×C271—21no—
80C22×C4×C271—21no—
81C22×C4×C271—21no—
82C22×C4×C271—21no—
83C22×C4×C271—21no—
84C22×C4×C271—21no—
85C22×C4×C271—21no—
86C22×C4×C271—21no—
87C22×C4×C271—21no—
88C22×C4×C271—21no—
89C22×C4×C271C2×C412yes—
90C22×C4×C271D412yes—
91C22×C4×C271D412yes—
92C22×D271—22no—
93C22×D271—22no—
94C22×D271—22no—
95C22×D271—22no—
96C22×D271—22no—
97C22×D271—22no—
98C22×D271—22no—
99C22×D271—22no—
100C22×D271—22no—
101C22×D271—22no—
102C22×D271—22no—
103C22×D271—22no—
104C22×D271—22no—
105C22×D271—22no—
106C22×D271—22no—
107C22×D271—22no—
108C22×D271—22no—
109C22×D271—22no—
110C22×D271—41no—
111C22×D271—41no—
112C22×D271—41no—
113C22×D271—41no—
114C22×D271—41no—
115C22×D271—41no—
116C22×D271—41no—
117C22×D271—41no—
118C22×D271—41no—
119C22×D271—41no—
120C22×D271—41no—
121C22×D271—41no—
122C22×D271—41no—
123C22×D271—41no—
124C22×D271—41no—
125C22×D271—41no—
126C22×D271C4⋊C414yescomplemented
127C2×Dic271—22no—
128C2×Dic271—22no—
129C2×Dic271—22no—
130C2×Dic271—22no—
131C2×Dic271—22no—
132C2×Dic271—22no—
133C2×Dic271—22no—
134C2×Dic271—22no—
135C2×Dic271—22no—
136C2×Dic271—22no—
137C2×Dic271—22no—
138C2×Dic271—22no—
139C2×Dic271—41no—
140C2×Dic271—41no—
141C2×Dic271—41no—
142C2×Dic271—41no—
143C2×Dic271—41no—
144C2×Dic271—41no—
145C2×Dic271—41no—
146C2×Dic271—41no—
147C2×Dic271C22⋊C414yescomplemented
148C23×C271C22×C411yescharacteristic
149C23×C271C2×D411yescharacteristic
150C23×C271C2×D411yescharacteristic
151C23×C271C2×D411yescharacteristic
152C23×C271C2×D411yescharacteristic
153C23×C271C2×D411yescharacteristic
154C23×C271C2×D411yescharacteristic
155C23×C271C2×D411yescharacteristic
156C23×C271C2×Q811yescharacteristic
157C23×C271#4111yescharacteristic
158C23×C271#4111yescharacteristic
159C23×C271#4111yescharacteristic
160C23×C271#4111yescharacteristic
161C23×C271#4111yescharacteristic
162C23×C271#4111yescharacteristic
163C2×C4×C271—22no—
164C2×C4×C271—22no—
165C2×C4×C271—22no—
166C2×C4×C271—22no—
167C2×C4×C271—22no—
168C2×C4×C271—22no—
169C2×C4×C271—22no—
170C2×C4×C271—22no—
171C2×C4×C271—22no—
172C2×C4×C271—41no—
173C2×C4×C271—41no—
174C2×C4×C271—41no—
175C2×C4×C271—41no—
176C2×C4×C271—41no—
177C2×C4×C271—41no—
178C2×C4×C271—41no—
179C2×C4×C271—41no—
180C2×C4×C271—41no—
181C2×C4×C271—41no—
182C2×C4×C271—41no—
183C2×C4×C271—41no—
184C22×C271C2×C22⋊C411yescharacteristic
185C22×C271C2×C4⋊C411yescharacteristic
186C22×C271#11711yescharacteristic
187C22×C271C4×D411yescharacteristic
188C22×C271C4×D411yescharacteristic
189C22×C271C4×D411yescharacteristic
190C22×C271C4×D411yescharacteristic
191C22×C271#12011yescharacteristic
192C22×C271#12011yescharacteristic
193C22×C271#12011yescharacteristic
194C22×C271#12111yescharacteristic
195C22×C271#12111yescharacteristic
196C22×C271#12111yescharacteristic
197C22×C271#12111yescharacteristic
198C22×C271#12111yescharacteristic
199C22×C271#12111yescharacteristic
200C22×C271#12111yescharacteristic
201C22×C271#12111yescharacteristic
202C22×C271C22⋊Q811yescharacteristic
203C22×C271C22⋊Q811yescharacteristic
204C22×C271C22⋊Q811yescharacteristic
205C22×C271C22⋊Q811yescharacteristic
206C22×C271C22⋊Q811yescharacteristic
207C22×C271C22⋊Q811yescharacteristic
208C22×C271#12311yescharacteristic
209C22×C271#12311yescharacteristic
210C22×C271#12311yescharacteristic
211C22×C271#12311yescharacteristic
212C22×C271#12311yescharacteristic
213C22×C271#12411yescharacteristic
214C22×C271#12411yescharacteristic
215C22×C271#12411yescharacteristic
216C22×C271#12611yescharacteristic
217C22×C271#12611yescharacteristic
218C22×C271#12711yescharacteristic
219C2×D271—24no—
220C2×D271—24no—
221C2×D271—24no—
222C2×D271—42no—
223C2×D271—42no—
224C2×D271—42no—
225C2×D271—42no—
226C2×D271—42no—
227C2×D271—42no—
228C2×D271—42no—
229C2×D271—42no—
230C2×D271—42no—
231C2×D271—42no—
232C2×D271—42no—
233C2×D271—42no—
234C4×C271—42no—
235C4×C271—42no—
236C4×C271—42no—
237Dic271—24no—
238Dic271—42no—
239Dic271—42no—
240C2×C271#38011yescharacteristic
241C2×C271#38511yescharacteristic
242C2×C271#38511yescharacteristic
243C2×C271#38611yescharacteristic
244C2×C271#38811yescharacteristic
245C2×C271#38811yescharacteristic
246C2×C271#39011yescharacteristic
247C2×C271#39211yescharacteristic
248C2×C271#39311yescharacteristic
249C2×C271#39411yescharacteristic
250C2×C271#39411yescharacteristic
251C2×C271#39611yescharacteristic
252C2×C271#39611yescharacteristic
253C2×C271#39711yescharacteristic
254C2×C271#39911yescharacteristic
255D271—44no—
256C271#143911yescharacteristic, complemented, Sylow
257#1439—2711nomaximal, Sylow
258C22×C4⋊C4—2711no—
259C22×C22⋊C4—2711no—
260C22×C22⋊C4—2711no—
261C22×C22⋊C4—2711no—
262C2×#95—2711no—
263C2×#95—2711no—
264C2×#95D27111yescharacteristic, complemented
265C25—2711no—
266C23×C4—2711no—
267C23×C4—2711no—
268C23×C4—2711no—
269C23×C4C2×D27111yescharacteristic
270C23×C4C2×D27111yescharacteristic
271C23×C4C2×D27111yescharacteristic
272C2×C4⋊C4—5422no—
273C2×C4⋊C4—5422no—
274C2×C4⋊C4—5422no—
275C2×C22⋊C4—2714no—
276C2×C22⋊C4—5422no—
277C2×C22⋊C4—5422no—
278C2×C22⋊C4—5422no—
279C2×C22⋊C4—5422no—
280C2×C22⋊C4—5422no—
281C2×C22⋊C4—5422no—
282C2×C22⋊C4—5422no—
283C2×C22⋊C4—5422no—
284#95—22no—
285#95—5422no—
286#95—5422no—
287C24—2712no—
288C24—2712no—
289C24—2712no—
290C24—5421no—
291C24—5421no—
292C24—5421no—
293C24—5421no—
294C24—5421no—
295C24—5421no—
296C24—5421no—
297C24—5421no—
298C24—5421no—
299C24—5421no—
300C24—5421no—
301C24—5421no—
302C24C22×D27111yescharacteristic
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—2712no—
322C22×C4—2712no—
323C22×C4—2712no—
324C22×C4—2712no—
325C22×C4—2712no—
326C22×C4—5421no—
327C22×C4—5421no—
328C22×C4—5421no—
329C22×C4—5421no—
330C22×C4—5421no—
331C22×C4—5421no—
332C22×C4—5421no—
333C22×C4—5421no—
334C22×C4—5421no—
335C22×C4—5421no—
336C22×C4—5421no—
337C22×C4—5421no—
338C22×C4—5421no—
339C22×C4—5421no—
340C22×C4—5421no—
341C22×C4—5421no—
342C22×C4C4×D27112yes—
343C22×C4C22⋊D27112yes—
344C22×C4D108412yes—
345C4⋊C4—5428no—
346C22⋊C4—5428no—
347C22⋊C4—5428no—
348C22⋊C4—5428no—
349C23—2714no—
350C23—5422no—
351C23—5422no—
352C23—5422no—
353C23—5422no—
354C23—5422no—
355C23—5422no—
356C23—5422no—
357C23—5422no—
358C23—5422no—
359C23—5422no—
360C23—5422no—
361C23—5422no—
362C23—5422no—
363C23—5422no—
364C23—5422no—
365C23—5422no—
366C23—5422no—
367C23—5422no—
368C23—10841no—
369C23—10841no—
370C23—10841no—
371C23—10841no—
372C23—10841no—
373C23—10841no—
374C23—10841no—
375C23—10841no—
376C23—10841no—
377C23—10841no—
378C23—10841no—
379C23—10841no—
380C23—10841no—
381C23—10841no—
382C23—10841no—
383C23—10841no—
384C23C2×C4×D27111yescharacteristic
385C23D4×D27111yescharacteristic
386C23D4×D27111yescharacteristic
387C23D4×D27111yescharacteristic
388C23D4×D27111yescharacteristic
389C23D4×D27111yescharacteristic
390C23C2×C22⋊D27111yescharacteristic
391C23C2×D108411yescharacteristic
392C23Q8×D27111yescharacteristic
393C23#1686465311yescharacteristic
394C23#1686465411yescharacteristic
395C23#1686465411yescharacteristic
396C23#1686465411yescharacteristic
397C23#1686465411yescharacteristic
398C23#1686465511yescharacteristic
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—2714no—
421C2×C4—5422no—
422C2×C4—5422no—
423C2×C4—5422no—
424C2×C4—5422no—
425C2×C4—5422no—
426C2×C4—5422no—
427C2×C4—5422no—
428C2×C4—5422no—
429C2×C4—5422no—
430C2×C4—5422no—
431C2×C4—5422no—
432C2×C4—5422no—
433C2×C4—10841no—
434C2×C4—10841no—
435C2×C4—10841no—
436C2×C4—10841no—
437C2×C4—10841no—
438C2×C4—10841no—
439C2×C4—10841no—
440C2×C4—10841no—
441C22—5424no—
442C22—5424no—
443C22—5424no—
444C22—10842no—
445C22—10842no—
446C22—10842no—
447C22—10842no—
448C22—10842no—
449C22—10842no—
450C22—10842no—
451C22—10842no—
452C22—10842no—
453C22—10842no—
454C22—10842no—
455C22—10842no—
456C22C22⋊C4×D27111yescharacteristic
457C22C4⋊C4×D27111yescharacteristic
458C22#2257611711yescharacteristic
459C22(C22⋊D271)⋊C411yescharacteristic
460C22C4×C22⋊D27111yescharacteristic
461C22#2257612211yescharacteristic
462C22C4×D108411yescharacteristic
463C22#2257612811yescharacteristic
464C22#2257613011yescharacteristic
465C22#2257613011yescharacteristic
466C22#2257613211yescharacteristic
467C22#2257613311yescharacteristic
468C22#2257613311yescharacteristic
469C22#2257613311yescharacteristic
470C22#2257613311yescharacteristic
471C22#2257613311yescharacteristic
472C22#2257613411yescharacteristic
473C22#2257613511yescharacteristic
474C22#2257613611yescharacteristic
475C22#2257613811yescharacteristic
476C22#2257613811yescharacteristic
477C22#2257613811yescharacteristic
478C22#2257613811yescharacteristic
479C22#2257614011yescharacteristic
480C22#2257614211yescharacteristic
481C22#2257614311yescharacteristic
482C22#2257614511yescharacteristic
483C22#2257614511yescharacteristic
484C22#2257614511yescharacteristic
485C22#2257614611yescharacteristic
486C22#2257614611yescharacteristic
487C22#2257614811yescharacteristic
488C22#2257615411yescharacteristic
489C22#2257615511yescharacteristic
490C22#2257615611yescharacteristic
491C4—42no—
492C4—42no—
493C4—42no—
494C4—5424no—
495C4—10842no—
496C4—10842no—
497C2—10844no—
498C2#3177375611yescharacteristic
499C2#3177376811yescharacteristic
500C2#3177377111yescharacteristic
501C2#3177377511yescharacteristic
502C2#3177378611yescharacteristic
503C2#3177378811yescharacteristic
504C2#3177379711yescharacteristic
505C2#3177380611yescharacteristic
506C2#3177381011yescharacteristic
507C2#3177381211yescharacteristic
508C2#3177381511yescharacteristic
509C2#3177381911yescharacteristic
510C2#3177382111yescharacteristic
511C2#3177382611yescharacteristic
512C2#3177383211yescharacteristic
subgroupG/H
center, FrattiniC24C22×D271
commutatorC23×C271C22×C4
FittingC2×#95×C271C2
socle, cosocleC24×C271C23
prime radicalC24×D271C22
hypercenterC2×#95D271
hypocenterC271#1439
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C24 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C22×Dic271⋊C4C2semidirect
5. C2×C271⋊#95C2semidirect
6. C2×C271⋊#95C2semidirect
7. C2×#95×C271C2semidirect
23. C2×(C2×D271)⋊C4C4semidirect
126. C22×D271C4⋊C4semidirect
147. C2×Dic271C22⋊C4semidirect
256. C271#1439semidirect
264. C2×#95D271semidirect

Maximal subgroups

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

subgroupindex
C22×Dic271⋊C42
C22×(C2×D271)⋊C42
C2×C271⋊#952
C2×#95×C2712
#1439271