Finite Group Databasebeta

← #239224

Subgroups — #239224

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1542C211yesmaximal, characteristic
7#10541C214yesmaximal, complemented
8#10538C214yesmaximal, complemented
9#7877C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1989—22no—
24#1988—22no—
25#1988C2212yes—
26#1988C2214yes—
27#1987C2212yes—
28#1939—22no—
29#1939C2214yes—
30#1797—22no—
31#1736—21no—
32#1736—21no—
33C2×#340—21no—
34C2×#340—21no—
35#1549—24no—
36#1542—24no—
37C2×#333C2212yes—
38C2×#332C2212yes—
39C2×#332C2214yes—
40#1277C2214yes—
41C23×Q8—21no—
42C22×Q16—24no—
43C22×M16—21no—
44C23×C8—21no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533D414yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8C2312yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#494—24no—
72#494—24no—
73#487—24no—
74#487—24no—
75#462—216no—
76#451—216no—
77#431—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8D412yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—24no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8D412yes—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4C2312yes—
101C2×#95—21no—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—22no—
107#340—22no—
108#336—24no—
109#333—24no—
110#332—24no—
111#332—28no—
112C22×Q8—24no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—41no—
117C22×Q8—42no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—41no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4#4111yescharacteristic
126C2×Q16—216no—
127C2×Q16—84no—
128C2×Q16—84no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8C2×D414yes—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—21no—
143C4⋊Q8—22no—
144C4⋊Q8—21no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8C2×D414yes—
155C4⋊Q8C2×D412yes—
156C4⋊Q8C2×D412yes—
157#125—24no—
158#125—42no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—42no—
166C4×Q8—42no—
167#117—24no—
168#117—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4Q1614yescomplemented
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—41no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42—82no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42#4111yescharacteristic
203C2×C42C2411yescharacteristic
204C4⋊C8—24no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—84no—
214#103—216no—
215C22⋊C8—24no—
216C22⋊C8—42no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—42no—
220#95—22no—
221#95—22no—
222C24#12111yescharacteristic
223C2×Q8—24no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—44no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—88no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×Q8—88no—
238C2×Q8—81no—
239C2×Q8—81no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4#12012yes—
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12112yes—
262C22×C4#12112yes—
263C22×C4#12111yescharacteristic
264C22×C4#12312yes—
265C22×C4#12312yes—
266C22×C4#12412yes—
267C22×C4C2×Q1612yes—
268C22×C4#13512yes—
269C22×C4#13712yes—
270C22×C4#13712yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274Q16—816no—
275M16—84no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—84no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4C2×Q1614yes—
311C22⋊C4—44no—
312C42—44no—
313C42—44no—
314C42—82no—
315C42#12012yes—
316C42#12112yes—
317C42#12112yes—
318C42#12312yes—
319C42#12312yes—
320C42#12412yes—
321C42C22×D411yescharacteristic
322C42C22×D411yescharacteristic
323C42C2×#4111yescharacteristic
324C42C2×#4111yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#39412yes—
330C23#46712yes—
331C23#47012yes—
332C23C2×#12111yescharacteristic
333Q8—44no—
334Q8—48no—
335Q8—82no—
336Q8—82no—
337Q8—88no—
338Q8—82no—
339Q8—84no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—88no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#39412yes—
369C2×C4#39412yes—
370C2×C4#39412yes—
371C2×C4#46214yes—
372C2×C4#46214yes—
373C2×C4#46414yes—
374C2×C4#46414yes—
375C2×C4#46712yes—
376C2×C4#47012yes—
377C2×C4C2×#12011yescharacteristic
378C2×C4C2×#12111yescharacteristic
379C2×C4C2×#12111yescharacteristic
380C2×C4C2×#12311yescharacteristic
381C2×C4C2×#12311yescharacteristic
382C2×C4C2×#12411yescharacteristic
383C2×C4C22×Q1611yescharacteristic
384C2×C4C2×#13511yescharacteristic
385C2×C4C2×#13711yescharacteristic
386C2×C4C2×#13711yescharacteristic
387C2×C4#57612yes—
388C2×C4#57712yes—
389C8—44no—
390C8—82no—
391C8—82no—
392C8—82no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22#205614yes—
400C22C2×#39411yescharacteristic
401C22#304111yescharacteristic
402C22#304311yescharacteristic
403C22C2×#46711yescharacteristic
404C22C2×#47011yescharacteristic
405C22#306911yescharacteristic
406C22#306911yescharacteristic
407C4—22no—
408C4—21no—
409C4—21no—
410C4—22no—
411C4—44no—
412C4—84no—
413C4—82no—
414C4—82no—
415C4—82no—
416C4#205614yes—
417C4#301212yes—
418C4C2×#46211yescharacteristic
419C4C2×#46211yescharacteristic
420C4C2×#46411yescharacteristic
421C4C2×#46411yescharacteristic
422C4#313612yes—
423C2—24no—
424C2#1938711yescharacteristic
425C2#1939911yescharacteristic
426C2#1942311yescharacteristic
427C2#1943511yescharacteristic
428C2#2351111yescharacteristic
429C2#2352211yescharacteristic
430C2C2×#205611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
7. #10541C2semidirect
8. #10538C2semidirect
188. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#219512
#213122
C2×#19882
C2×#15492
C2×#15422
#105412
#105382
#78772