Finite Group Databasebeta

← #116103

Subgroups — #116103

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1986C211yesmaximal, characteristic
2C2×#1515C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4#13964C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×#383C2211yescharacteristic
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1986—22no—
16#1986C414yescomplemented
17C2×#339—21no—
18C2×#339—21no—
19#1515—22no—
20#1515—22no—
21#1506—22no—
22#1506—22no—
23C2×#328—22no—
24C2×#328—22no—
25#1441—22no—
26#1436—22no—
27C4⋊#95—21no—
28#1434—21no—
29#1300—22no—
30#1288—22no—
31#1285—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533D412yes—
41C2×C4⋊Q8Q811yescharacteristic
42C2×C4⋊Q8Q811yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C42—21no—
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432D412yes—
62#432D412yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16C2×C412yes—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×C4×C8—21no—
75#383—21no—
76#383—21no—
77#383—21no—
78#383—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2×C412yes—
82C4×C4⋊C4D412yes—
83#376—21no—
84#376—21no—
85#376—21no—
86#376—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95D412yes—
95C2×#95D412yes—
96#339—24no—
97#339—24no—
98#328—28no—
99#328—28no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×M16—24no—
110C2×M16—22no—
111C2×M16—42no—
112C2×M16—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8C4⋊C412yescomplemented
123C4⋊Q8C4⋊C412yescomplemented
124#125—22no—
125#125—22no—
126#125—41no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144#117—22no—
145#117—22no—
146#117—41no—
147#117—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—82no—
161C2×C22⋊C4—41no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—44no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×Q811yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—41no—
196#97—41no—
197#97—41no—
198C4×C8—22no—
199C4×C8—41no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—44no—
207#95—44no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—82no—
215C24#12011yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—81no—
221C2×Q8—81no—
222C2×Q8—82no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12411yescharacteristic
269C22×C4#12411yescharacteristic
270C22×C4#12411yescharacteristic
271C22×C4#12711yescharacteristic
272M16—24no—
273M16—42no—
274M16—44no—
275C2×C8—44no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—81no—
280C2×C8—81no—
281C2×C8—82no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—88no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42—42no—
300C42—42no—
301C42—44no—
302C42—44no—
303C42—42no—
304C42—42no—
305C42—81no—
306C42—81no—
307C42C2×C4⋊C411yescharacteristic
308C42#11711yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42#12111yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C42#12311yescharacteristic
321C42#12611yescharacteristic
322C42#12611yescharacteristic
323C23—21no—
324C23—21no—
325C23—22no—
326C23#38611yescharacteristic
327C23#39211yescharacteristic
328C23#39311yescharacteristic
329C23#45512yes—
330C23#45612yes—
331Q8—82no—
332Q8—82no—
333Q8—82no—
334Q8—82no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#38011yescharacteristic
387C2×C4#38511yescharacteristic
388C2×C4#38511yescharacteristic
389C2×C4#38811yescharacteristic
390C2×C4#38811yescharacteristic
391C2×C4#39011yescharacteristic
392C2×C4#39411yescharacteristic
393C2×C4#39411yescharacteristic
394C2×C4#39611yescharacteristic
395C2×C4#39611yescharacteristic
396C2×C4#39711yescharacteristic
397C2×C4#39911yescharacteristic
398C8—44no—
399C8—82no—
400C8—82no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22#143911yescharacteristic
410C22#188511yescharacteristic
411C22#188611yescharacteristic
412C22#200712yes—
413C22#202812yes—
414C22#203111yescharacteristic
415C22#203111yescharacteristic
416C22#203112yes—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—44no—
430C4—82no—
431C4—84no—
432C4—82no—
433C4—82no—
434C4—162no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2#1037912yes—
439C2#1038112yes—
440C2#1105511yescharacteristic
441C2#1108111yescharacteristic
442C2#1108511yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
16. #1986C4semidirect
122. C4⋊Q8C4⋊C4semidirect
123. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19862
C2×#15152
C2×#15062
#139642
C2×#13002
C2×#12882
C2×#12852