Finite Group Databasebeta

← #496408

Subgroups — #496408

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#20728C211yesmaximal, characteristic
4#13302C214yesmaximal, complemented
5#13301C214yesmaximal, complemented
6#12863C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9#2920—21no—
10#2920—21no—
11#2857—21no—
12#2857—21no—
13#2628—21no—
14#2628—21no—
15#2568C2212yes—
16#2567C2212yes—
17#2563C412yes—
18#2542C412yes—
19#2487—21no—
20#2487—21no—
21#2409—21no—
22#2409—21no—
23#2317C412yes—
24#2311C2212yes—
25#2285C411yescharacteristic
26#1822—21no—
27#1822—21no—
28#1801C2212yes—
29#1801C2212yes—
30#1800—21no—
31#1800—21no—
32#1785C2212yes—
33#1785C2212yes—
34C4×#101C2212yes—
35C4×#101C2212yes—
36C4×#100—21no—
37C4×#100—21no—
38#1394C22116yes—
39C22×#41C2311yescharacteristic
40C23×Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C2×C4×Q8C2311yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C412yes—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C412yes—
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72#432—41no—
73#432—41no—
74#432—41no—
75#432—41no—
76#431—41no—
77#431—41no—
78#431—41no—
79#431—41no—
80#423—22no—
81#423—22no—
82C2×#101C2314yes—
83C2×#101C2314yes—
84C2×#100—22no—
85C2×#100—22no—
86#407—22no—
87#407—22no—
88C4×M16—22no—
89C4×M16—22no—
90#400—22no—
91#400—22no—
92#399—22no—
93#399—22no—
94#395—22no—
95#395—22no—
96#393—22no—
97#393—22no—
98#391—22no—
99#391—22no—
100#391C2314yes—
101#390C2×C414yes—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#386—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#384—21no—
111#384C2×C414yes—
112#382—21no—
113#382C2×C414yes—
114#380—22no—
115#380—22no—
116#380—21no—
117#380—21no—
118#380—22no—
119#380—22no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122#343—22no—
123#343—22no—
124#343—22no—
125#343—22no—
126C2×#41—24no—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—42no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—84no—
135C22×Q8C2411yescharacteristic
136C22×D4C22×C411yescharacteristic
137C22×D4C2×D411yescharacteristic
138C22×D4C2×D411yescharacteristic
139C23×C4—22no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C22⋊Q8—84no—
162C22⋊Q8—84no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179#117—41no—
180#117—41no—
181#117—41no—
182#117—41no—
183#117—41no—
184#117—41no—
185#117—44no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C22×C414yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4C22⋊C411yescharacteristic
209C2×C22⋊C4C22⋊C411yescharacteristic
210C2×C22⋊C4C22⋊C411yescharacteristic
211C2×C22⋊C4C22⋊C411yescharacteristic
212C2×C22⋊C4C22⋊C411yescharacteristic
213C2×C22⋊C4C22⋊C411yescharacteristic
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C414yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241#101—28no—
242#101—28no—
243#100—44no—
244#100—44no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249#97—42no—
250#97—42no—
251C4×C8—42no—
252C4×C8—42no—
253#95—22no—
254#95—22no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259C24C2×C22⋊C411yescharacteristic
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C411yescharacteristic
262#41—48no—
263#41—48no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—84no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×Q82−1+414yes—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—84no—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330M16—44no—
331M16—44no—
332M16—44no—
333M16—44no—
334C2×C8—44no—
335C2×C8—44no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—44no—
360C22⋊C4—84no—
361C22⋊C4—84no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—44no—
381C42#9911yescharacteristic
382C42#9911yescharacteristic
383C42#9911yescharacteristic
384C42#9911yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23#41012yes—
400C23C22×C22⋊C411yescharacteristic
401Q8—44no—
402Q8—84no—
403Q8—88no—
404Q8—84no—
405D4—44no—
406D4—44no—
407D4—44no—
408C2×C4—24no—
409C2×C4—24no—
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—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—88no—
455C2×C4C2×#9911yescharacteristic
456C2×C4C2×#9911yescharacteristic
457C2×C4C2×#9911yescharacteristic
458C2×C4C2×#9911yescharacteristic
459C2×C4C2×#9911yescharacteristic
460C2×C4C2×#9911yescharacteristic
461C2×C4#51912yes—
462C2×C4#53612yes—
463C2×C4#53612yes—
464C8—84no—
465C8—84no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—81no—
482C22—81no—
483C22#231011yescharacteristic
484C22C22×#9911yescharacteristic
485C22C2×#41011yescharacteristic
486C4—22no—
487C4—22no—
488C4—21no—
489C4—21no—
490C4—22no—
491C4—21no—
492C4—21no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—84no—
500C4—84no—
501C4#213412yes—
502C4#213512yes—
503C2—21no—
504C2—21no—
505C2—42no—
506C2—81no—
507C2—81no—
508C2#2182511yescharacteristic
509C2C2×#213411yescharacteristic
510C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13302C2semidirect
5. #13301C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#207282
#133022
#133012
#128632