Finite Group Databasebeta

← #117783

Subgroups — #117783

Order 512.

664 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21894C211yesmaximal, characteristic
2C2×#1848C211yesmaximal, characteristic
3C2×#1561C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5C2×#1309C211yesmaximal, characteristic
6#8041C211yesmaximal, characteristic
7#7897C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×#108—22no—
10C2×#423C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2285C2211yescharacteristic
16#1988—21no—
17#1981—21no—
18#1894—22no—
19#1848—22no—
20#1848—22no—
21#1803—22no—
22#1561—22no—
23#1561—22no—
24C2×#332—21no—
25C2×#332—21no—
26#1493—22no—
27#1493C414yescomplemented
28C2×Q8⋊C8—22no—
29#1309—22no—
30#1296—22no—
31#1279—22no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16D412yes—
36C23×C8—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533Q812yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55C2×#108—28no—
56C2×#108—48no—
57#423—21no—
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#423—21no—
69#423—21no—
70#423C2×C412yes—
71#423D412yes—
72#423D412yes—
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⋊C8—21no—
81C2×#103—24no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2×C412yes—
86C2×C4×C8D412yes—
87C2×C4×C8D412yes—
88#384D412yes—
89#382—21no—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94#340—24no—
95#332—24no—
96#332—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—44no—
112C2×M16—44no—
113C2×M16—81no—
114C2×M16—82no—
115C2×M16—81no—
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C4⋊C412yes—
132C4⋊Q8C4⋊C412yes—
133#125—22no—
134#125—22no—
135#125—41no—
136#125—41no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—22no—
154#117—22no—
155#117—41no—
156#117—41no—
157#117—81no—
158#117—81no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×Q811yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184#108—416no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201#103—84no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8C22⋊C414yes—
206#95—41no—
207#95—41no—
208C24#12411yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4#11711yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12411yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12611yescharacteristic
248M16—44no—
249M16—44no—
250M16—82no—
251M16—82no—
252M16—82no—
253M16—82no—
254M16—82no—
255M16—82no—
256C2×C8—24no—
257C2×C8—28no—
258C2×C8—28no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—82no—
267C2×C8—81no—
268C2×C8—82no—
269C2×C8—81no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—82no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—82no—
290C42—42no—
291C42—42no—
292C42—81no—
293C42—81no—
294C42C2×C22⋊C411yescharacteristic
295C42C2×C4⋊C411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12011yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42#12311yescharacteristic
307C42#12311yescharacteristic
308C42#12611yescharacteristic
309C42#12711yescharacteristic
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23#38811yescharacteristic
315C23#39311yescharacteristic
316C23#39411yescharacteristic
317Q8—42no—
318Q8—42no—
319Q8—44no—
320Q8—81no—
321Q8—81no—
322Q8—81no—
323Q8—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#38011yescharacteristic
357C2×C4#38511yescharacteristic
358C2×C4#38511yescharacteristic
359C2×C4#38611yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#39011yescharacteristic
362C2×C4#39211yescharacteristic
363C2×C4#39411yescharacteristic
364C2×C4#39611yescharacteristic
365C2×C4#39611yescharacteristic
366C2×C4#39711yescharacteristic
367C2×C4#39911yescharacteristic
368C2×C4#45312yes—
369C2×C4#45612yes—
370C2×C4#47111yescharacteristic
371C2×C4#47111yescharacteristic
372C2×C4#47112yes—
373C2×C4#47211yescharacteristic
374C2×C4#47311yescharacteristic
375C2×C4#47312yes—
376C8—28no—
377C8—82no—
378C8—82no—
379C8—82no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22#143911yescharacteristic
390C22#203912yes—
391C22#205111yescharacteristic
392C22#205212yes—
393C22#205211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—82no—
402C4—162no—
403C4#181511yescharacteristic
404C4#181611yescharacteristic
405C4#185912yes—
406C4#197711yescharacteristic
407C4#197911yescharacteristic
408C4#200211yescharacteristic
409C4#200511yescharacteristic
410C4#205111yescharacteristic
411C4#205211yescharacteristic
412C4#206412yes—
413C2—41no—
414C2—41no—
415C2#1053212yes—
416C2#1053412yes—
417C2#1107811yescharacteristic
418C2#1110511yescharacteristic
419C2#1112811yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
27. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#218942
C2×#18482
C2×#15612
C2×#14932
C2×#13092
#80412
#78972