Finite Group Databasebeta

← #121986

Subgroups — #121986

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2#10834C211yesmaximal, characteristic
3#7908C211yesmaximal, characteristic
4#7891C212yesmaximal
5#7890C212yesmaximal
6C2×#544C411yescharacteristic
7C2×#431C2212yes—
8C22×C4⋊C8C2211yescharacteristic
9C22×C4×C8C2211yescharacteristic
10#2331—22no—
11#2330—22no—
12#2329—22no—
13#2317C412yes—
14#2316C411yescharacteristic
15C4×C4⋊Q8—22no—
16C4×#125—22no—
17C4×C22⋊Q8—22no—
18C4×#117C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#336—22no—
21#1280—22no—
22#1279—22no—
23#1279—22no—
24#1279—22no—
25#1279—22no—
26C23×C8—21no—
27C23×C8—21no—
28#544—24no—
29#544—22no—
30#544C814yescomplemented
31C2×C4⋊Q8—22no—
32C2×#125—22no—
33C2×C22⋊Q8—22no—
34C2×C4×Q8—22no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C42D412yes—
40C22×C42C2311yescharacteristic
41#431—22no—
42#431—22no—
43#431—22no—
44#431Q814yescomplemented
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8Q812yescomplemented
50C2×C22⋊C8—22no—
51C2×C4×C8—22no—
52C2×C4×C8—22no—
53C2×C4×C8—22no—
54C2×C4×C8—21no—
55C2×C4×C8Q812yescomplemented
56#391—24no—
57#391—22no—
58#389—24no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389—22no—
63#389—22no—
64#387—24no—
65#387—24no—
66#387—24no—
67#385—24no—
68#384—22no—
69#382—22no—
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4D412yes—
76C4×C4⋊C4D412yes—
77C4×C4⋊C4D412yes—
78C4×C4⋊C4D412yes—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81#376—22no—
82C2×#95D412yes—
83C43—22no—
84#336—44no—
85C22×Q8—22no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8—41no—
103C22×C8—41no—
104C4⋊Q8—28no—
105C4⋊Q8—28no—
106#125—28no—
107#125—28no—
108C22⋊Q8—28no—
109C22⋊Q8—28no—
110C4×Q8—82no—
111C4×Q8—82no—
112#117—22no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117C2×C814yes—
119#117M1614yes—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C2×C812yes—
137C2×C4⋊C4M1612yes—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42C2×Q811yescharacteristic
153C2×C42C2×Q811yescharacteristic
154C2×C42C2×Q812yes—
155C2×C42#4112yes—
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C4⋊C8—24no—
159C4⋊C8—24no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162C22⋊C8—24no—
163C22⋊C8—42no—
164C4×C8—24no—
165C4×C8—24no—
166C4×C8—42no—
167C4×C8—42no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176C24C2×C22⋊C411yescharacteristic
177C2×Q8—28no—
178C2×Q8—82no—
179C2×Q8—82no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C22⋊C814yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C4×Q811yescharacteristic
202C22×C4C4×Q811yescharacteristic
203C22×C4C4×Q812yes—
204C22×C4#12011yescharacteristic
205C22×C4#12012yes—
206C22×C4#12011yescharacteristic
207C22×C4C22⋊Q811yescharacteristic
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q812yes—
210C22×C4#12411yescharacteristic
211C22×C4#12411yescharacteristic
212C22×C4C4⋊Q811yescharacteristic
213C22×C4C4⋊Q811yescharacteristic
214C22×C4C22×C811yescharacteristic
215C22×C4C2×M1612yes—
216C22×C4C2×M1611yescharacteristic
217C22×C4#13111yescharacteristic
218C22×C4#13111yescharacteristic
219C22×C4#13112yes—
220C2×C8—24no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—42no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—81no—
234C2×C8—81no—
235C4⋊C4—28no—
236C4⋊C4—216no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—84no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—84no—
251C4⋊C4C22⋊C814yes—
252C4⋊C4C22⋊C814yes—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C42—24no—
258C42—24no—
259C42—42no—
260C42—42no—
261C42—42no—
262C42—42no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—82no—
268C42—82no—
269C42—82no—
270C42C22⋊Q812yes—
271C42C22⋊Q812yes—
272C42C22⋊Q812yes—
273C42C22⋊Q812yes—
274C42#12412yes—
275C42#12412yes—
276C42C4⋊Q812yes—
277C42C4⋊Q812yes—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#37911yescharacteristic
283C23#39111yescharacteristic
284C23#39111yescharacteristic
285C23C2×C22⋊C811yescharacteristic
286C23#40711yescharacteristic
287C23#40811yescharacteristic
288C23#40811yescharacteristic
289Q8—88no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#39112yes—
328C2×C4#39112yes—
329C2×C4#39311yescharacteristic
330C2×C4#39312yes—
331C2×C4#39311yescharacteristic
332C2×C4#39511yescharacteristic
333C2×C4#39512yes—
334C2×C4#39511yescharacteristic
335C2×C4C2×C22⋊C812yes—
336C2×C4#40712yes—
337C2×C4#40812yes—
338C2×C4#40812yes—
339C2×C4C8×Q811yescharacteristic
340C2×C4C8×Q811yescharacteristic
341C2×C4C8×Q812yes—
342C2×C4#44612yes—
343C2×C4#44612yes—
344C2×C4#44612yes—
345C2×C4#44612yes—
346C2×C4#44611yescharacteristic
347C2×C4#44611yescharacteristic
348C2×C4#44612yes—
349C2×C4#45311yescharacteristic
350C2×C4#45411yescharacteristic
351C2×C4#45511yescharacteristic
352C2×C4#45611yescharacteristic
353C8—44no—
354C8—84no—
355C8—82no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22#144611yescharacteristic
369C22#177911yescharacteristic
370C22#178011yescharacteristic
371C22#179211yescharacteristic
372C22#179311yescharacteristic
373C22#179411yescharacteristic
374C22#179411yescharacteristic
375C22#198411yescharacteristic
376C22#198411yescharacteristic
377C22#198511yescharacteristic
378C22#198511yescharacteristic
379C4—21no—
380C4—21no—
381C4—22no—
382C4—22no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4#198412yes—
390C4#198412yes—
391C4#198512yes—
392C4#198512yes—
393C4#202711yescharacteristic
394C4#202811yescharacteristic
395C4#206011yescharacteristic
396C4#206111yescharacteristic
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2#831811yescharacteristic
402C2#831911yescharacteristic
403C2#832011yescharacteristic
404C2#832111yescharacteristic
405C2#1138011yescharacteristic
406C2#1139211yescharacteristic
407C2#1139311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#379
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #544C8semidirect
44. #431Q8semidirect
49. C2×C4⋊C8Q8semidirect
55. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
#108342
#79082
#78912
#78902