Finite Group Databasebeta

← #117935

Subgroups — #117935

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21312C211yesmaximal, characteristic
2C2×#1567C211yesmaximal, characteristic
3C2×#1550C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#10847C211yesmaximal, characteristic
6C2×#1292C211yesmaximal, characteristic
7C2×#1279C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#423C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1989—22no—
16#1988—21no—
17#1987—21no—
18C2×#343—22no—
19#1736—22no—
20#1567—22no—
21#1567—22no—
22C2×#335—22no—
23#1550—22no—
24#1550—22no—
25C2×#333—21no—
26C2×#332—21no—
27#1493—22no—
28#1493C414yescomplemented
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31#1292—22no—
32#1279—21no—
33#1279—21no—
34C22×M16—21no—
35C22×M16—21no—
36C22×M16—21no—
37C23×C8—21no—
38C23×C8—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533C2×C412yes—
46#533Q812yes—
47C2×C4⋊Q8D411yescharacteristic
48C2×C4⋊Q8D411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#117—21no—
55C22×C4⋊C4D412yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#423—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8D412yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2×C412yes—
86C2×C4×C8D412yes—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C2×#95—21no—
94#343—44no—
95#340—24no—
96#340—24no—
97#335—28no—
98#333—24no—
99#332—24no—
100Q8⋊C8—24no—
101Q8⋊C8—24no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4—22no—
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—42no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—44no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—81no—
132C22×C8—81no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8C4⋊C412yes—
140C4⋊Q8C4⋊C412yes—
141#125—22no—
142#125—22no—
143#125—41no—
144#125—41no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161#117—22no—
162#117—22no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4Q1614yes—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—81no—
182C2×C42—81no—
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q811yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8C22⋊C412yes—
216C4×C8C22⋊C412yes—
217#95—41no—
218#95—41no—
219C24#12311yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—82no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4#11711yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12611yescharacteristic
258C22×C4#12711yescharacteristic
259C22×C4C2×Q1612yes—
260C22×C4#13512yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263M16—44no—
264M16—44no—
265M16—82no—
266M16—81no—
267M16—81no—
268M16—81no—
269M16—81no—
270M16—81no—
271M16—81no—
272C2×C8—24no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—44no—
283C2×C8—82no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—81no—
287C2×C8—81no—
288C2×C8—82no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C42—42no—
308C42—42no—
309C42—82no—
310C42—81no—
311C42—81no—
312C42C2×C22⋊C411yescharacteristic
313C42C2×C4⋊C411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42#12011yescharacteristic
317C42#12011yescharacteristic
318C42#12111yescharacteristic
319C42#12111yescharacteristic
320C42#12111yescharacteristic
321C42#12111yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42#12311yescharacteristic
326C42#12411yescharacteristic
327C42#12611yescharacteristic
328C23—22no—
329C23—21no—
330C23—21no—
331C23#38811yescharacteristic
332C23#39411yescharacteristic
333C23#39911yescharacteristic
334C23#48412yes—
335C23#48512yes—
336Q8—42no—
337Q8—42no—
338Q8—82no—
339Q8—82no—
340Q8—81no—
341Q8—81no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—84no—
368C2×C4#38011yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4#38511yescharacteristic
371C2×C4#38611yescharacteristic
372C2×C4#38811yescharacteristic
373C2×C4#39011yescharacteristic
374C2×C4#39211yescharacteristic
375C2×C4#39311yescharacteristic
376C2×C4#39411yescharacteristic
377C2×C4#39611yescharacteristic
378C2×C4#39611yescharacteristic
379C2×C4#39711yescharacteristic
380C2×C4C4×Q1612yes—
381C2×C4#44012yes—
382C2×C4#44912yes—
383C2×C4#45112yes—
384C2×C4#46212yes—
385C2×C4#46312yes—
386C2×C4#46712yes—
387C2×C4#46812yes—
388C2×C4#47111yescharacteristic
389C2×C4#47111yescharacteristic
390C2×C4#47211yescharacteristic
391C2×C4#47311yescharacteristic
392C2×C4#47712yes—
393C2×C4#47912yes—
394C2×C4#49411yescharacteristic
395C2×C4#49511yescharacteristic
396C2×C4#49611yescharacteristic
397C2×C4#49711yescharacteristic
398C8—24no—
399C8—42no—
400C8—82no—
401C8—82no—
402C8—82no—
403C8—82no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22#143911yescharacteristic
413C22#192812yes—
414C22#203612yes—
415C22#205012yes—
416C22#205111yescharacteristic
417C22#205211yescharacteristic
418C22#209511yescharacteristic
419C22#209711yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—82no—
427C4—82no—
428C4—84no—
429C4#181511yescharacteristic
430C4#181611yescharacteristic
431C4#186312yes—
432C4#196911yescharacteristic
433C4#197311yescharacteristic
434C4#201211yescharacteristic
435C4#201611yescharacteristic
436C4#205111yescharacteristic
437C4#205211yescharacteristic
438C4#206812yes—
439C2—21no—
440C2—21no—
441C2—22no—
442C2#1046112yes—
443C2#1046312yes—
444C2#1108211yescharacteristic
445C2#1110511yescharacteristic
446C2#1112411yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ 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
28. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#213122
C2×#15672
C2×#15502
C2×#14932
#108472
C2×#12922
C2×#12792