Finite Group Databasebeta

← #16793360

Subgroups — #16793360

Order 3328.

1214 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131512C211yesmaximal, characteristic
2#13131512C211yesmaximal, characteristic
3#13130178C214yesmaximal, complemented
4#13130164C214yesmaximal, complemented
5#13130138C214yesmaximal, complemented
6C22×Q8×Dic13C211yesmaximal, characteristic
7C23×Q8×C13C411yescharacteristic
8C2×Q8×Dic13—21no—
9C2×Q8×Dic13—21no—
10C2×Q8×Dic13—21no—
11C2×Q8×Dic13—21no—
12C2×Q8×Dic13—21no—
13C2×Q8×Dic13—21no—
14C2×Q8×Dic13C412yes—
15C2×Q8×Dic13C412yes—
16C2×Q8×Dic13C412yes—
17C2×Q8×Dic13C2212yes—
18C2×Q8×Dic13C2212yes—
19C2×Q8×Dic13C2212yes—
20C22×C4⋊Dic13—21no—
21C22×C4⋊Dic13C411yescharacteristic
22C22×C4×Dic13—21no—
23C22×C4×Dic13C2211yescharacteristic
24#11719169—21no—
25#11719169—21no—
26#11719130—21no—
27#11719130C2212yes—
28#11719129—21no—
29#11719129C2212yes—
30#11719128—21no—
31#11719128—21no—
32#11719125—21no—
33#11719125C2212yes—
34C4×#6122—21no—
35C4×#6122C2212yes—
36#11719119C2212yes—
37#11719119C2212yes—
38Q8⋊#1057C22116yescomplemented
39C22×Q8×C13—28no—
40C22×Q8×C13—21no—
41C22×Q8×C13—21no—
42C22×Q8×C13—21no—
43C22×Q8×C13C2×C412yes—
44C22×Q8×C13C2×C412yes—
45C22×Q8×C13C2×C412yes—
46C23×C4×C13—21no—
47C23×C4×C13C2×C411yescharacteristic
48C23×Dic13—21no—
49C23×Dic13D411yescharacteristic
50C23×Dic13D411yescharacteristic
51C2×#6122—22no—
52C2×#6122—22no—
53Q8×Dic13—22no—
54Q8×Dic13—22no—
55Q8×Dic13—22no—
56Q8×Dic13—22no—
57Q8×Dic13—22no—
58Q8×Dic13—22no—
59Q8×Dic13—22no—
60Q8×Dic13—22no—
61Q8×Dic13—22no—
62Q8×Dic13—22no—
63Q8×Dic13—22no—
64Q8×Dic13—22no—
65Q8×Dic13C2×C414yes—
66Q8×Dic13C2×C414yes—
67Q8×Dic13C2×C414yes—
68Q8×Dic13C2314yes—
69C2×C4⋊Dic13—21no—
70C2×C4⋊Dic13—21no—
71C2×C4⋊Dic13—21no—
72C2×C4⋊Dic13—22no—
73C2×C4⋊Dic13—22no—
74C2×C4⋊Dic13—22no—
75C2×C4⋊Dic13—22no—
76C2×C4⋊Dic13—22no—
77C2×C4⋊Dic13—22no—
78C2×C4⋊Dic13C2×C412yes—
79C2×C4⋊Dic13C2×C412yes—
80C2×C4⋊Dic13C2×C412yes—
81C2×C4×Dic13—22no—
82C2×C4×Dic13—22no—
83C2×C4×Dic13—22no—
84C2×C4×Dic13—22no—
85C2×C4×Dic13—22no—
86C2×C4×Dic13—22no—
87C2×C4×Dic13C2×C411yescharacteristic
88C2×C4×Dic13C2×C411yescharacteristic
89C2×C4×Dic13C2×C411yescharacteristic
90C2×C4×Dic13D411yescharacteristic
91C2×C4×Dic13D411yescharacteristic
92C2×C4×Dic13D411yescharacteristic
93C2×C4×Dic13D411yescharacteristic
94C2×C4×Dic13D411yescharacteristic
95C2×C4×Dic13D411yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C22⋊#1057—22no—
100C22⋊#1057—22no—
101C22⋊#1057—22no—
102C22⋊#1057—22no—
103#88112—22no—
104#88112—22no—
105#88110—22no—
106#88110—22no—
107C4⋊#1057—22no—
108C4⋊#1057C2314yes—
109C4×#1057—22no—
110C4×#1057C2314yes—
111#17449—131nomaximal, Sylow
112C24×C13C22⋊C411yescharacteristic
113C2×Q8×C13—28no—
114C2×Q8×C13—28no—
115C2×Q8×C13—28no—
116C2×Q8×C13—28no—
117C2×Q8×C13—28no—
118C2×Q8×C13—22no—
119C2×Q8×C13—22no—
120C2×Q8×C13—22no—
121C2×Q8×C13—24no—
122C2×Q8×C13M1618yescomplemented
123C2×Q8×C13C22×C414yes—
124C22×C4×C13—28no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—24no—
129C22×C4×C13C22⋊C411yescharacteristic
130C22×C4×C13C22⋊C411yescharacteristic
131C22×C4×C13C22⋊C411yescharacteristic
132C22×C4×C13C22×C411yescharacteristic
133C22×C4×C13C22×C411yescharacteristic
134C22×C4×C13C22×C411yescharacteristic
135C22×Dic13—24no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—42no—
140C22×Dic13—44no—
141C22×Dic13C22⋊C411yescharacteristic
142C22×Dic13C22⋊C411yescharacteristic
143C22×Dic13C22⋊C411yescharacteristic
144C22×Dic13C22⋊C411yescharacteristic
145C22×Dic13C22⋊C411yescharacteristic
146C22×Dic13C22⋊C411yescharacteristic
147C22×Dic13C2×D411yescharacteristic
148C22×Dic13C2×D411yescharacteristic
149C22×Dic13C2×D411yescharacteristic
150C22×Dic13C2×D411yescharacteristic
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153#6122—44no—
154#6122—44no—
155C2×#1057—24no—
156C2×#1057—24no—
157C4⋊Dic13—22no—
158C4⋊Dic13—22no—
159C4⋊Dic13—22no—
160C4⋊Dic13—24no—
161C4⋊Dic13—24no—
162C4⋊Dic13—24no—
163C4⋊Dic13—24no—
164C4⋊Dic13C22×C414yes—
165C4×Dic13—24no—
166C4×Dic13—24no—
167C4×Dic13—24no—
168C4×Dic13—24no—
169C4×Dic13C22⋊C411yescharacteristic
170C4×Dic13C22⋊C411yescharacteristic
171C4×Dic13C22⋊C411yescharacteristic
172C4×Dic13C22⋊C411yescharacteristic
173C4×Dic13C22⋊C411yescharacteristic
174C4×Dic13C22⋊C411yescharacteristic
175C4×Dic13C22×C411yescharacteristic
176C4×Dic13C22×C411yescharacteristic
177C4×Dic13C22×C411yescharacteristic
178C4×Dic13C2×D411yescharacteristic
179C4×Dic13C2×D411yescharacteristic
180C4×Dic13C2×D411yescharacteristic
181C4×Dic13C2×D411yescharacteristic
182C4×Dic13C2×D411yescharacteristic
183C4×Dic13C2×D411yescharacteristic
184C4×Dic13C2411yescharacteristic
185C22×C4×Q8—131no—
186#2920—131no—
187#2920—131no—
188#1491—134no—
189#1487—134no—
190#1479—134no—
191C23×C13—22no—
192C23×C13—22no—
193C23×C13—22no—
194C23×C13C2×C22⋊C411yescharacteristic
195C23×C13C2×C22⋊C411yescharacteristic
196C23×C13C2×C22⋊C411yescharacteristic
197Q8×C13—28no—
198Q8×C13—28no—
199Q8×C13—24no—
200Q8×C13—28no—
201Q8×C13C2×M1618yes—
202C2×C4×C13—28no—
203C2×C4×C13—28no—
204C2×C4×C13—24no—
205C2×C4×C13—28no—
206C2×C4×C13—22no—
207C2×C4×C13—22no—
208C2×C4×C13—22no—
209C2×C4×C13—22no—
210C2×C4×C13—22no—
211C2×C4×C13C2×C22⋊C411yescharacteristic
212C2×C4×C13C2×C22⋊C411yescharacteristic
213C2×C4×C13C2×C22⋊C411yescharacteristic
214C2×C4×C13C2×M1614yes—
215C2×C4×C13C23×C411yescharacteristic
216C2×Dic13—22no—
217C2×Dic13—22no—
218C2×Dic13—22no—
219C2×Dic13—22no—
220C2×Dic13—22no—
221C2×Dic13—22no—
222C2×Dic13—22no—
223C2×Dic13—24no—
224C2×Dic13—41no—
225C2×Dic13—41no—
226C2×Dic13—41no—
227C2×Dic13—41no—
228C2×Dic13—44no—
229C2×Dic13—44no—
230C2×Dic13—44no—
231C2×Dic13C2×C22⋊C411yescharacteristic
232C2×Dic13C2×C22⋊C411yescharacteristic
233C2×Dic13C2×C22⋊C411yescharacteristic
234C2×Dic13C2×C22⋊C411yescharacteristic
235C2×Dic13C2×C22⋊C411yescharacteristic
236C2×Dic13C2×C22⋊C411yescharacteristic
237C2×Dic13#13714yes—
238C2×Dic13C22×D411yescharacteristic
239C2×Dic13C22×D411yescharacteristic
240#1057—44no—
241#1057—44no—
242C23×Q8#26111yescharacteristic
243C2×C4×Q8—132no—
244C2×C4×Q8—132no—
245C2×C4×Q8—132no—
246C2×C4×Q8—132no—
247C2×C4×Q8—132no—
248C2×C4×Q8—132no—
249C2×C4×Q8—261no—
250C2×C4×Q8—261no—
251C2×C4×Q8—261no—
252C2×C4×Q8—261no—
253C2×C4×Q8—261no—
254C2×C4×Q8—261no—
255C22×C4⋊C4—131no—
256C22×C4⋊C4—261no—
257C22×C42—131no—
258C22×C42—261no—
259#432—132no—
260#432—261no—
261#432—261no—
262#432—261no—
263#431—132no—
264#431—132no—
265#431—132no—
266#431—261no—
267#423—132no—
268#423—261no—
269#407—261no—
270#407—261no—
271C4×M16—132no—
272C4×M16—261no—
273Q8⋊C8—1316no—
274C22×C13—28no—
275C22×C13—21no—
276C22×C13—21no—
277C22×C13—21no—
278C22×C13—21no—
279C22×C13—21no—
280C22×C13—21no—
281C22×C13#40712yes—
282C22×C13#41912yes—
283C22×C13#42112yes—
284C22×C13C22×C22⋊C411yescharacteristic
285C4×C13—28no—
286C4×C13—21no—
287C4×C13—21no—
288C4×C13#40712yes—
289C4×C13C22×M1612yes—
290Dic13—21no—
291Dic13—21no—
292Dic13—21no—
293Dic13—21no—
294Dic13—21no—
295Dic13—21no—
296Dic13—44no—
297Dic13#41912yes—
298Dic13C2×#13712yes—
299C22×Q8—28no—
300C22×Q8—21no—
301C22×Q8—21no—
302C22×Q8—21no—
303C22×Q8C2×#26112yes—
304C22×Q8C2×#26112yes—
305C22×Q8C2×#26112yes—
306C23×C4—21no—
307C23×C4—131no—
308C23×C4—131no—
309C23×C4—261no—
310C23×C4C2×#26111yescharacteristic
311C2×M16—262no—
312C2×M16—262no—
313C4×Q8—134no—
314C4×Q8—134no—
315C4×Q8—134no—
316C4×Q8—134no—
317C4×Q8—262no—
318C4×Q8—262no—
319C4×Q8—262no—
320C4×Q8—262no—
321C4×Q8—262no—
322C4×Q8—262no—
323C4×Q8—262no—
324C4×Q8—262no—
325C4×Q8—262no—
326C4×Q8—262no—
327C4×Q8—262no—
328C4×Q8—262no—
329C2×C4⋊C4—132no—
330C2×C4⋊C4—132no—
331C2×C4⋊C4—132no—
332C2×C4⋊C4—262no—
333C2×C4⋊C4—262no—
334C2×C4⋊C4—262no—
335C2×C4⋊C4—262no—
336C2×C4⋊C4—262no—
337C2×C4⋊C4—262no—
338C2×C4⋊C4—261no—
339C2×C4⋊C4—261no—
340C2×C4⋊C4—261no—
341C2×C42—131no—
342C2×C42—131no—
343C2×C42—131no—
344C2×C42—131no—
345C2×C42—131no—
346C2×C42—131no—
347C2×C42—131no—
348C2×C42—131no—
349C2×C42—131no—
350C2×C42—131no—
351C2×C42—131no—
352C2×C42—131no—
353C2×C42—262no—
354C2×C42—262no—
355C2×C42—262no—
356C2×C42—262no—
357C2×C42—262no—
358C2×C42—262no—
359C4⋊C8—134no—
360C4⋊C8—262no—
361C4⋊C8—262no—
362C4⋊C8—262no—
363C22⋊C8—262no—
364C22⋊C8—262no—
365C22⋊C8—262no—
366C22⋊C8—262no—
367#97—262no—
368#97—262no—
369C4×C8—134no—
370C4×C8—262no—
371C2×C13—22no—
372C2×C13—22no—
373C2×C13—22no—
374C2×C13C2×#40711yescharacteristic
375C2×C13C2×#41911yescharacteristic
376C2×C13C2×#42111yescharacteristic
377C24C22⋊#26111yescharacteristic
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8#612218yescomplemented
388C2×Q8C22×#26114yes—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—131no—
395C22×C4—131no—
396C22×C4—131no—
397C22×C4—131no—
398C22×C4—131no—
399C22×C4—131no—
400C22×C4—131no—
401C22×C4—131no—
402C22×C4—131no—
403C22×C4—131no—
404C22×C4—131no—
405C22×C4—131no—
406C22×C4—264no—
407C22×C4—262no—
408C22×C4—262no—
409C22×C4—262no—
410C22×C4—522no—
411C22×C4—524no—
412C22×C4C22⋊#26111yescharacteristic
413C22×C4C22⋊#26111yescharacteristic
414C22×C4C22⋊#26111yescharacteristic
415C22×C4C22×#26111yescharacteristic
416C22×C4C22×#26111yescharacteristic
417C22×C4C22×#26111yescharacteristic
418M16—524no—
419M16—524no—
420C2×C8—264no—
421C2×C8—264no—
422C4⋊C4—134no—
423C4⋊C4—264no—
424C4⋊C4—264no—
425C4⋊C4—264no—
426C4⋊C4—264no—
427C4⋊C4—262no—
428C4⋊C4—262no—
429C4⋊C4—262no—
430C42—131no—
431C42—131no—
432C42—131no—
433C42—131no—
434C42—131no—
435C42—131no—
436C42—131no—
437C42—131no—
438C42—131no—
439C42—131no—
440C42—131no—
441C42—131no—
442C42—131no—
443C42—131no—
444C42—131no—
445C42—131no—
446C42—264no—
447C42—264no—
448C42—264no—
449C42—264no—
450C13#1744911yescharacteristic, complemented, Sylow
451C23—22no—
452C23—22no—
453C23—22no—
454C23C2×C22⋊#26111yescharacteristic
455C23C2×C22⋊#26111yescharacteristic
456C23C2×C22⋊#26111yescharacteristic
457Q8—28no—
458Q8—28no—
459Q8—28no—
460Q8—24no—
461Q8C2×#612218yes—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—131no—
472C2×C4—131no—
473C2×C4—134no—
474C2×C4—131no—
475C2×C4—131no—
476C2×C4—131no—
477C2×C4—131no—
478C2×C4—131no—
479C2×C4—131no—
480C2×C4—262no—
481C2×C4—262no—
482C2×C4—262no—
483C2×C4—262no—
484C2×C4—262no—
485C2×C4—262no—
486C2×C4—262no—
487C2×C4—264no—
488C2×C4—521no—
489C2×C4—521no—
490C2×C4—521no—
491C2×C4—521no—
492C2×C4—524no—
493C2×C4—524no—
494C2×C4—524no—
495C2×C4C2×#612214yes—
496C2×C4C2×C22⋊#26111yescharacteristic
497C2×C4C2×C22⋊#26111yescharacteristic
498C2×C4C2×C22⋊#26111yescharacteristic
499C2×C4C23×#26111yescharacteristic
500C8—524no—
501C8—524no—
502C22—28no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22#1171915312yes—
510C22#1171915512yes—
511C22#1171916912yes—
512C22C22×C22⋊#26111yescharacteristic
513C4—28no—
514C4—21no—
515C4—21no—
516C4—132no—
517C4—132no—
518C4—261no—
519C4—261no—
520C4—261no—
521C4—261no—
522C4—261no—
523C4—261no—
524C4—524no—
525C4#1171916912yes—
526C4C22×#612212yes—
527C2—22no—
528C2—22no—
529C2—22no—
530C2C2×#1171916911yescharacteristic
531C2C2×#1171915311yescharacteristic
532C2C2×#1171915511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
FittingC23×Q8×C13C4
socleC22×C13C22×C22⋊C4
prime radicalC24×C13C22⋊C4
cosocleC4×Dic13C24
hypercenterC23×Q8#261
hypocenterC13#17449
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #13130178C2semidirect
4. #13130164C2semidirect
5. #13130138C2semidirect
38. Q8⋊#1057C22semidirect
122. C2×Q8×C13M16semidirect
387. C2×Q8#6122semidirect
450. C13#17449semidirect

Maximal subgroups

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

subgroupindex
#131315122
#131301782
#131301642
#131301382
C22×Q8×Dic132
#1744913