Finite Group Databasebeta

← #106969

Subgroups — #106969

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2C2×#1805C211yesmaximal, characteristic
3C2×#1491C211yesmaximal, characteristic
4#8033C211yesmaximal, characteristic
5C2×#1290C211yesmaximal, characteristic
6C2×#1283C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—22no—
10#2984—22no—
11#2984—22no—
12#2984C414yescomplemented
13C2×#446—22no—
14C2×#446—21no—
15C2×C8×Q8—21no—
16C2×#432—21no—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431—21no—
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103C412yes—
23C2×C4×M16C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#1805—21no—
26#1805—21no—
27#1805—21no—
28#1805—21no—
29C2×#340C412yes—
30#1491—22no—
31#1491C414yescomplemented
32C2×Q8⋊C8—21no—
33C2×Q8⋊C8—21no—
34#1295—22no—
35#1290—22no—
36#1283—22no—
37#1280—21no—
38#1280—21no—
39C23×Q8C2×C411yescharacteristic
40C22×M16—21no—
41C23×C8C2×C412yes—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8Q812yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#446—28no—
62#446—24no—
63#446—24no—
64#446—28no—
65C8×Q8—24no—
66C8×Q8—24no—
67#432—24no—
68#432—21no—
69#432—21no—
70#432—24no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432C2×C412yes—
75#432C2×C412yes—
76#432D412yes—
77#431—24no—
78#431—21no—
79#431—21no—
80#431—24no—
81#431C2×C412yes—
82#431D412yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—44no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101C4×M16—21no—
102C4×M16—21no—
103C4×M16—21no—
104C4×M16C2×C412yes—
105C2×#97—22no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97D412yes—
109C2×C4×C8—22no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112#382—21no—
113#382—21no—
114#382—21no—
115#382—21no—
116C2×#95—21no—
117C2×#95—21no—
118#340—22no—
119#340—22no—
120Q8⋊C8—24no—
121Q8⋊C8—24no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8C4212yes—
130C22×Q8C22⋊C411yescharacteristic
131C22×Q8C22⋊C411yescharacteristic
132C22×Q8C4⋊C412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C2×M16—24no—
140C2×M16—22no—
141C2×M16—42no—
142C2×M16—41no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—42no—
152C22×C8—42no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—21no—
158C4×Q8—22no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—22no—
169C4×Q8C22⋊C412yes—
170C4×Q8C22⋊C412yes—
171C4×Q8C22⋊C412yes—
172C4×Q8C22⋊C412yes—
173C4×Q8C4⋊C412yes—
174C4×Q8C4⋊C412yes—
175C4×Q8C4⋊C412yes—
176C4×Q8C4⋊C412yes—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C4212yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C4⋊C412yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q811yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C4⋊C8—28no—
217C4⋊C8—28no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227C4⋊C8—41no—
228#103—44no—
229#103—44no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—22no—
235C22⋊C8—41no—
236C22⋊C8C22⋊C412yes—
237C22⋊C8C22⋊C412yes—
238#97—28no—
239#97—22no—
240#97—22no—
241#97—22no—
242#97—41no—
243#97—41no—
244#97C22⋊C414yes—
245C4×C8—28no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—22no—
249C4×C8—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254C24C4×D411yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8#9512yes—
276C2×Q8#9512yes—
277C2×Q8#10012yes—
278C2×Q8#10112yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C2×C4211yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4#11711yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4C22⋊Q811yescharacteristic
318C22×C4C22⋊Q811yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4#12311yescharacteristic
321M16—24no—
322M16—44no—
323M16—42no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—82no—
337C2×C8—82no—
338C2×C8—82no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—82no—
354C4⋊C4#9512yes—
355C4⋊C4#9512yes—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42C2×C22⋊C411yescharacteristic
365C42C2×C22⋊C411yescharacteristic
366C42C2×C22⋊C411yescharacteristic
367C42C2×C4⋊C411yescharacteristic
368C42C2×C4⋊C411yescharacteristic
369C42#11711yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42#12011yescharacteristic
373C42#12011yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42C22⋊Q811yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42#12311yescharacteristic
379C42#12311yescharacteristic
380C23—21no—
381C23—21no—
382C23—22no—
383C23C4×C22⋊C411yescharacteristic
384C23#38511yescharacteristic
385C23#38611yescharacteristic
386C23#44312yes—
387C23#44412yes—
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8—22no—
392Q8—22no—
393Q8—44no—
394Q8—44no—
395Q8—41no—
396Q8—41no—
397Q8#34314yes—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
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—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—162no—
441C2×C4—162no—
442C2×C4C2×#9511yescharacteristic
443C2×C4C4×C22⋊C411yescharacteristic
444C2×C4#37911yescharacteristic
445C2×C4#38011yescharacteristic
446C2×C4#38011yescharacteristic
447C2×C4#38111yescharacteristic
448C2×C4#38511yescharacteristic
449C2×C4#38511yescharacteristic
450C2×C4#38511yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4C2×#10011yescharacteristic
455C2×C4C2×#10111yescharacteristic
456C2×C4#41311yescharacteristic
457C2×C4#41311yescharacteristic
458C8—42no—
459C8—42no—
460C8—44no—
461C8—44no—
462C8—82no—
463C8—82no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22C22⋊#9511yescharacteristic
473C22#176411yescharacteristic
474C22#176511yescharacteristic
475C22#186612yes—
476C22#187612yes—
477C22#190011yescharacteristic
478C22#190111yescharacteristic
479C22#190512yes—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—24no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—162no—
495C4C2×#34311yescharacteristic
496C4#174211yescharacteristic
497C4#190011yescharacteristic
498C4#190111yescharacteristic
499C2—22no—
500C2—21no—
501C2—21no—
502C2#924412yes—
503C2#924612yes—
504C2#1073911yescharacteristic
505C2#1078611yescharacteristic
506C2#1079111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2984C4semidirect
31. #1491C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
C2×#18052
C2×#14912
#80332
C2×#12902
C2×#12832
#79142