Finite Group Databasebeta

← #97584

Subgroups — #97584

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3#8311C212yesmaximal
4C4×#324C211yesmaximal, characteristic
5#7575C212yesmaximal
6C22×C4⋊Q8C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2365—24no—
10#2365—24no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#389—21no—
14C2×#387—21no—
15C2×#387—21no—
16C2×C4×C4⋊C4—21no—
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×Q8⋊C8—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1272—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8C2×C411yescharacteristic
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8C814yescomplemented
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—44no—
39C2×C4×Q8—44no—
40C2×C4×Q8D414yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C4×C8—22no—
59#391—22no—
60#391—22no—
61#391—22no—
62#391D414yes—
63#389—28no—
64#387—28no—
65#387—28no—
66C4×C4⋊C4—28no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95D411yescharacteristic
77C2×#95D411yescharacteristic
78#336—22no—
79#336—22no—
80Q8⋊C8—48no—
81#324—24no—
82#324—42no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8—21no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—42no—
92C22×Q8—42no—
93C22×Q8C22⋊C412yes—
94C22×Q8C22⋊C412yes—
95C22×Q8C2×C812yes—
96C22×Q8C2×C812yes—
97C22×Q8M1612yes—
98C22×Q8M1612yes—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C22×C8—42no—
111C22×C8—42no—
112C4⋊Q8—28no—
113C4⋊Q8—28no—
114C4⋊Q8—28no—
115C4⋊Q8—28no—
116C4×Q8—28no—
117C4×Q8—44no—
118C4×Q8—48no—
119C4×Q8—48no—
120C4×Q8—44no—
121C4×Q8—44no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4SD1612yes—
144C2×C4⋊C4Q1612yes—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—24no—
154C2×C42—42no—
155C2×C42—44no—
156C2×C42—44no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42C2×C812yes—
161C2×C42M1612yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C4⋊C8—44no—
165C22⋊C8—24no—
166C22⋊C8—42no—
167C22⋊C8—42no—
168C22⋊C8—42no—
169C4×C8—44no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—22no—
177C24C2×C22⋊C411yescharacteristic
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8C22⋊C814yes—
201C2×Q8C22⋊C814yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4C22⋊C812yes—
248C22×C4C22⋊C812yes—
249C22×C4#10312yes—
250C22×C4#10312yes—
251C22×C4#10412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4#12012yes—
255C22×C4#12012yes—
256C22×C4C22×C811yescharacteristic
257C22×C4C2×M1611yescharacteristic
258C22×C4C2×M1611yescharacteristic
259C22×C4C2×M1611yescharacteristic
260C22×C4C2×SD1611yescharacteristic
261C22×C4C2×Q1611yescharacteristic
262C22×C4#13712yes—
263C2×C8—44no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C2×C8—82no—
268C2×C8—82no—
269C4⋊C4—24no—
270C4⋊C4—216no—
271C4⋊C4—216no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—44no—
276C4⋊C4—48no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4#9914yes—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—24no—
286C42—42no—
287C42—48no—
288C42—48no—
289C42—42no—
290C42—48no—
291C42—42no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23#37911yescharacteristic
297C23C2×C22⋊C811yescharacteristic
298C23#40711yescharacteristic
299C23C2×#10311yescharacteristic
300C23#41911yescharacteristic
301C23C2×#10411yescharacteristic
302C23#42111yescharacteristic
303Q8—28no—
304Q8—44no—
305Q8—44no—
306Q8—44no—
307Q8—44no—
308Q8—44no—
309Q8—44no—
310Q8—44no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4Q8⋊C814yescomplemented
369C2×C4C2×C22⋊C811yescharacteristic
370C2×C4C2×C22⋊C811yescharacteristic
371C2×C4#40711yescharacteristic
372C2×C4#40711yescharacteristic
373C2×C4C2×#9912yes—
374C2×C4#41312yes—
375C2×C4#44812yes—
376C2×C4#45112yes—
377C8—84no—
378C8—162no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22C2×Q8⋊C811yescharacteristic
392C22#147911yescharacteristic
393C22#148711yescharacteristic
394C22#149111yescharacteristic
395C22#177911yescharacteristic
396C22#178111yescharacteristic
397C22#178511yescharacteristic
398C22#178711yescharacteristic
399C22#178911yescharacteristic
400C22#179611yescharacteristic
401C22#179611yescharacteristic
402C4—21no—
403C4—21no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—44no—
408C4—44no—
409C4—42no—
410C4—42no—
411C4—44no—
412C4—42no—
413C4—81no—
414C4—82no—
415C4#145912yes—
416C4#146512yes—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2#830111yescharacteristic
422C2#831111yescharacteristic
423C2#832311yescharacteristic
424C2#836411yescharacteristic
425C2#836811yescharacteristic
426C2#837911yescharacteristic
427C2#839211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
34. C2×C4⋊Q8C8semidirect
368. C2×C4Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
#85582
#83112
C4×#3242
#75752