Finite Group Databasebeta

← #193302

Subgroups — #193302

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34371C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic
3#21963C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7899C211yesmaximal, characteristic
11#7894C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2381—22no—
17#2365C2212yes—
18#2365C2212yes—
19#2345—22no—
20Q8×C22⋊C4—22no—
21#2331—21no—
22#2331—21no—
23#2327—22no—
24#2313—22no—
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2283C2211yescharacteristic
28#1990C2212yes—
29#1990C2212yes—
30#1987—22no—
31#1949—22no—
32#1948C2212yes—
33#1948C2212yes—
34#1947C2212yes—
35#1947C2212yes—
36#1854—22no—
37#1846—21no—
38#1835—21no—
39#1824—22no—
40#1809—22no—
41#1546C418yescomplemented
42C2×#332C2212yes—
43C2×#332C2212yes—
44#1279C2211yescharacteristic
45#1279C2211yescharacteristic
46#1279C2211yescharacteristic
47#1279C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50C23×Q8—21no—
51C22×M16—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8Q814yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2×C414yes—
70C2×#107—22no—
71C2×#107—22no—
72C2×#106—22no—
73C2×#106—22no—
74#423—22no—
75#423C2×C414yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#391—24no—
87#391—24no—
88#391—22no—
89#391—22no—
90#389—22no—
91#389—22no—
92#387—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#384C2312yes—
99#383—21no—
100#383—21no—
101#382—21no—
102#382D412yes—
103#382D411yescharacteristic
104#382D411yescharacteristic
105#381—21no—
106#381—21no—
107#380—24no—
108#380—21no—
109#380—21no—
110#380D412yes—
111#380D412yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2312yes—
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—24no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122#376—21no—
123#376—21no—
124#340—22no—
125#340—22no—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#332C2×C418yes—
131#332C2×C418yes—
132C22×Q8—24no—
133C22×Q8—22no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—82no—
137C22×Q8—82no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—24no—
143C2×M16—82no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8C4⋊C418yes—
157C4⋊Q8C22×C414yes—
158C4⋊Q8C2×Q814yes—
159C4×Q8—82no—
160C4×Q8—82no—
161C4×Q8—82no—
162C4×Q8—82no—
163#117—81no—
164#117—81no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4C4⋊C418yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D414yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—41no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—41no—
191C2×C42C22×C412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×Q812yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C2×C42C2411yescharacteristic
199#107—82no—
200#107—82no—
201#106—82no—
202#106—82no—
203C4⋊C8—24no—
204C4⋊C8—22no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—42no—
208C4⋊C8C22×C414yes—
209C4⋊C8C22×C412yes—
210C4⋊C8C22×C412yes—
211C22⋊C8—24no—
212C22⋊C8—42no—
213C4×C8—24no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—41no—
234#95—41no—
235C24#12011yescharacteristic
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—28no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—84no—
244C2×Q8—88no—
245C2×Q8—84no—
246C2×Q8—84no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—84no—
262C22×C4C2×C4⋊C412yes—
263C22×C4C4×D412yes—
264C22×C4C4×D412yes—
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4#12312yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—84no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4C2×C4⋊C418yes—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—82no—
312C42—42no—
313C42—41no—
314C42—41no—
315C42—42no—
316C42—41no—
317C42—41no—
318C42—82no—
319C42—84no—
320C42—84no—
321C42C2×C4⋊C412yes—
322C42C4×D412yes—
323C42C4×D412yes—
324C42C22⋊Q812yes—
325C42C22⋊Q812yes—
326C42#12312yes—
327C42C23×C411yescharacteristic
328C42C22×Q811yescharacteristic
329C42C2×#4111yescharacteristic
330C42C2×#4111yescharacteristic
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23#38512yes—
336C23C2×#12011yescharacteristic
337Q8—44no—
338Q8—44no—
339Q8—44no—
340Q8—44no—
341Q8—88no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—28no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—88no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#38512yes—
376C2×C4#38512yes—
377C2×C4#38512yes—
378C2×C4C22×C4⋊C411yescharacteristic
379C2×C4C2×C4×D411yescharacteristic
380C2×C4C2×C4×D411yescharacteristic
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×#12311yescharacteristic
384C2×C4#57612yes—
385C2×C4#57712yes—
386C2×C4#57711yescharacteristic
387C2×C4#57711yescharacteristic
388C2×C4#57811yescharacteristic
389C2×C4#57811yescharacteristic
390C8—42no—
391C8—42no—
392C8—44no—
393C8—81no—
394C8—81no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—42no—
399C22—42no—
400C22C2×#38511yescharacteristic
401C22#301011yescharacteristic
402C22#301111yescharacteristic
403C22#301212yes—
404C22#302012yes—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—48no—
411C4—82no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—162no—
416C4#294711yescharacteristic
417C4#294711yescharacteristic
418C4#294811yescharacteristic
419C4#294911yescharacteristic
420C4#308112yes—
421C4#308211yescharacteristic
422C4#308211yescharacteristic
423C2—42no—
424C2#1894412yes—
425C2#1894512yes—
426C2#2085711yescharacteristic
427C2#2087711yescharacteristic
428C2#2087911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8925C2semidirect
8. #8923C2semidirect
9. #8922C2semidirect
41. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#343712
#219852
#219632
#218342
C2×#15462
#89262
#89252
#89232
#89222
#78992
#78942