Finite Group Databasebeta

← #196167

Subgroups — #196167

Order 512.

1069 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#17464C211yesmaximal, characteristic
4#17449C211yesmaximal, characteristic
5C2×#1461C211yesmaximal, characteristic
6#9092C214yesmaximal, complemented
7#9088C214yesmaximal, complemented
8#7921C211yesmaximal, characteristic
9#7863C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3113—22no—
13#3113—22no—
14#3113—22no—
15#3113—22no—
16C2×#483C412yes—
17C2×#475C412yes—
18C2×#462C412yes—
19C2×#451C412yes—
20#2986—22no—
21#2985—22no—
22#2984—22no—
23#2983C2212yes—
24#2982C2212yes—
25#2962—22no—
26#2960—22no—
27C2×#441C2214yes—
28#2924—22no—
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C2×#432C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34#1491—22no—
35#1487—22no—
36#1485—22no—
37#1481—22no—
38#1479C2214yes—
39#1479C2214yes—
40#1461—22no—
41#1461C2212yes—
42#1461C2212yes—
43C2×#324—22no—
44#1281C2212yes—
45#1281C2212yes—
46#1275C2214yes—
47C23×Q8C2×C411yescharacteristic
48C22×Q16C2×C414yes—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533D412yes—
54#533D412yes—
55#533D412yes—
56#533D412yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8C2312yes—
75C2×C4×Q8C2312yes—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#483—24no—
82#483—24no—
83#475—24no—
84#475—24no—
85#462—24no—
86#462—24no—
87#451—24no—
88#451—24no—
89#446—24no—
90#446—28no—
91#446—24no—
92#446—24no—
93C8×Q8—24no—
94C8×Q8—24no—
95C8×Q8—24no—
96#441—28no—
97#441—28no—
98#433—28no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432—24no—
106#432—21no—
107#432—21no—
108#431—21no—
109#431—21no—
110#431—24no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431D412yes—
116#431D412yes—
117#431C2312yes—
118#431C2312yes—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121#431C2312yes—
122C2×#106C2×C414yes—
123#424—24no—
124#423—21no—
125#423—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8C2×C412yes—
129C2×#103C2×C414yes—
130C2×#103C2×C414yes—
131#407—21no—
132#407—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8C2×C412yes—
135C2×C22⋊C8D412yes—
136#405—24no—
137C4×M16—21no—
138C4×M16D412yes—
139C2×#97C2312yes—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142Q8⋊C8—28no—
143Q8⋊C8—28no—
144#324—24no—
145#324—24no—
146C22×Q8—24no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—42no—
154C22×Q8C22×C412yes—
155C22×Q8C22×C412yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×Q16—28no—
162C2×Q16—28no—
163C2×Q16—44no—
164C2×M16—22no—
165C2×M16—41no—
166C22×C8—22no—
167C22×C8—41no—
168C22×C8C22×C414yes—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C22⋊C412yes—
174C4⋊Q8C22⋊C412yes—
175C4⋊Q8C22⋊C412yes—
176C4⋊Q8C22⋊C412yes—
177#125—22no—
178#125—22no—
179#125—22no—
180#125—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8C22⋊C414yes—
188C22⋊Q8C22⋊C414yes—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8C2×D414yes—
212C4×Q8C2×D414yes—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C22⋊C4—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42C2411yescharacteristic
255#106—28no—
256C4⋊C8—28no—
257C4⋊C8—28no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269#103—28no—
270#103—28no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—41no—
279C22⋊C8—41no—
280C22⋊C8C2×D412yes—
281C22⋊C8C2×D412yes—
282#97—28no—
283#97—22no—
284#97—22no—
285#97—22no—
286#97—41no—
287C4×C8—28no—
288C4×C8—22no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—41no—
294C4×C8C2×D414yes—
295C24C4×D411yescharacteristic
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—22no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—42no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—82no—
315C2×Q8C2×C22⋊C414yes—
316C2×Q8#13114yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—82no—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12311yescharacteristic
346C22×C4#12311yescharacteristic
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350Q16—416no—
351M16—44no—
352M16—82no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—44no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—82no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4C2×C22⋊C414yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C42—24no—
386C42—24no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42C2×C22⋊C411yescharacteristic
392C42C2×C22⋊C411yescharacteristic
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42#12011yescharacteristic
396C42#12011yescharacteristic
397C42#12111yescharacteristic
398C42#12111yescharacteristic
399C42#12111yescharacteristic
400C42#12111yescharacteristic
401C42#12311yescharacteristic
402C42#12311yescharacteristic
403C42C22×D411yescharacteristic
404C42C22×D411yescharacteristic
405C42C22×D411yescharacteristic
406C42C2×#4111yescharacteristic
407C23—22no—
408C23—21no—
409C23—21no—
410C23—41no—
411C23#38611yescharacteristic
412C23#38611yescharacteristic
413C23C2×C4×D411yescharacteristic
414Q8—24no—
415Q8—28no—
416Q8—24no—
417Q8—44no—
418Q8—44no—
419Q8—42no—
420Q8—44no—
421Q8—44no—
422Q8—42no—
423Q8—82no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4#38611yescharacteristic
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×#12011yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C2×C4C2×#13112yes—
469C2×C4#56812yes—
470C8—42no—
471C8—42no—
472C8—44no—
473C8—44no—
474C8—82no—
475C8—82no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22#190512yes—
486C22C2×#38611yescharacteristic
487C22#292911yescharacteristic
488C22#293011yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—44no—
499C4—44no—
500C4—44no—
501C4—82no—
502C4#190212yes—
503C4#288012yes—
504C4#293312yes—
505C2—22no—
506C2—21no—
507C2—21no—
508C2—41no—
509C2#2089911yescharacteristic
510C2C2×#190211yescharacteristic
511C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9092C2semidirect
7. #9088C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
#602502
#174642
#174492
C2×#14612
#90922
#90882
#79212
#78632