Finite Group Databasebeta

← #118363

Subgroups — #118363

Order 512.

717 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21941C211yesmaximal, characteristic
2#17578C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C2×#1472C211yesmaximal, characteristic
5#8107C211yesmaximal, characteristic
6C2×#1298C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1854—22no—
16#1853—21no—
17#1852—21no—
18#1808—21no—
19#1807—21no—
20#1520—22no—
21#1520—22no—
22#1520—21no—
23#1520—21no—
24#1511—22no—
25#1508—22no—
26C2×#328—22no—
27#1472—21no—
28#1472—21no—
29#1472C414yescomplemented
30C2×#324—22no—
31#1305—22no—
32#1298—22no—
33#1282—22no—
34C23×Q8Q811yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—21no—
37C22×M16—21no—
38C23×C8—21no—
39C23×C8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8D412yes—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4Q811yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C2×#107—22no—
59C2×#106—22no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66#407C2×C412yes—
67#407D412yes—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8C2×C412yes—
81C2×C22⋊C8D412yes—
82C2×C22⋊C8D412yes—
83C2×C22⋊C8D411yescharacteristic
84C2×C22⋊C8D411yescharacteristic
85#381—21no—
86#380—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90#380D411yescharacteristic
91#380D411yescharacteristic
92#380D412yes—
93#379—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C2×#95—21no—
97#340—22no—
98#340—22no—
99#328—28no—
100#328—24no—
101#328—24no—
102#324—24no—
103#324—24no—
104C25#4111yescharacteristic
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—82no—
111C22×Q8C4⋊C412yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—41no—
115C23×C4—82no—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q811yescharacteristic
125C23×C4#4111yescharacteristic
126C23×C4#4111yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C23×C4#4111yescharacteristic
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—41no—
133C2×M16—41no—
134C2×M16—82no—
135C2×M16—82no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—82no—
146C22×C8—82no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C22⋊Q8—41no—
161C22⋊Q8—41no—
162C22⋊Q8—41no—
163#117—82no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C4⋊C412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—82no—
194C2×C42—41no—
195#107—82no—
196#106—82no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8C22⋊C412yes—
214C22⋊C8C22⋊C412yes—
215#95—22no—
216#95—21no—
217#95—21no—
218#95—41no—
219#95—41no—
220C24—22no—
221C24—22no—
222C24—42no—
223C24—81no—
224C24#11711yescharacteristic
225C24#12111yescharacteristic
226C24#12111yescharacteristic
227C24#12111yescharacteristic
228C24C22⋊Q811yescharacteristic
229C24#12611yescharacteristic
230C24#12611yescharacteristic
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—82no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C4⋊C411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4C22⋊Q811yescharacteristic
286C22×C4C22⋊Q811yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12311yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4#12411yescharacteristic
295C22×C4#12411yescharacteristic
296C22×C4#12711yescharacteristic
297M16—82no—
298M16—82no—
299M16—82no—
300M16—82no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—82no—
309C2×C8—82no—
310C2×C8—82no—
311C2×C8—82no—
312C2×C8—82no—
313C2×C8—82no—
314C2×C8—82no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—82no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—82no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—84no—
362C23—84no—
363C23#38011yescharacteristic
364C23#38811yescharacteristic
365C23#38811yescharacteristic
366C23#39211yescharacteristic
367C23#39611yescharacteristic
368C23#39611yescharacteristic
369C23#39711yescharacteristic
370C23#47112yes—
371C23#47312yes—
372Q8—82no—
373Q8—82no—
374Q8—82no—
375Q8—82no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—164no—
422C2×C4#38511yescharacteristic
423C2×C4#38511yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#39011yescharacteristic
426C2×C4#39311yescharacteristic
427C2×C4#39411yescharacteristic
428C2×C4#39411yescharacteristic
429C2×C4#39911yescharacteristic
430C8—42no—
431C8—42no—
432C8—82no—
433C8—82no—
434C8—82no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22—81no—
459C22—81no—
460C22—82no—
461C22—82no—
462C22#143911yescharacteristic
463C22#181412yes—
464C22#193012yes—
465C22#201911yescharacteristic
466C22#202011yescharacteristic
467C22#206412yes—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—82no—
476C4—82no—
477C4—161no—
478C4—161no—
479C4—162no—
480C4#203212yes—
481C2—21no—
482C2—21no—
483C2—22no—
484C2—22no—
485C2—22no—
486C2—41no—
487C2—41no—
488C2—41no—
489C2#1108911yescharacteristic
490C2#1110411yescharacteristic
491C2#1114511yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1472C4semidirect

Maximal subgroups

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

subgroupindex
#219412
#175782
C2×#15202
C2×#14722
#81072
C2×#12982
#79272