Finite Group Databasebeta

← #196268

Subgroups — #196268

Order 512.

1097 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#35248C211yesmaximal, characteristic
3#17464C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9104C214yesmaximal, complemented
6#9100C214yesmaximal, complemented
7#7857C212yesmaximal
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—22no—
12#2983C2212yes—
13#2982C2212yes—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17C2×#431C2212yes—
18#2855—22no—
19#2634—22no—
20#2581—22no—
21#2568C411yescharacteristic
22#2568C411yescharacteristic
23#2546C411yescharacteristic
24#2546C411yescharacteristic
25#2503C412yes—
26#2484—22no—
27#2434C412yes—
28#2327C2212yes—
29#2327C2212yes—
30#2283C2211yescharacteristic
31#1485—24no—
32#1481—24no—
33#1479C2218yes—
34#1461—22no—
35#1461C2212yes—
36#1461C2212yes—
37C2×#324—22no—
38#1274C2218yes—
39C23×Q8C2×C411yescharacteristic
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8C2312yes—
61C2×C4×Q8C2311yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×#117—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4D412yes—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#446—216no—
69#446—28no—
70C8×Q8—28no—
71#432—24no—
72#432—24no—
73#432—22no—
74#432—22no—
75#431—24no—
76#431—24no—
77#431—22no—
78#431—22no—
79#431—22no—
80#431D414yes—
81#431C2314yes—
82#431C2314yes—
83#423—24no—
84#423—22no—
85C2×C4⋊C8—22no—
86#407—22no—
87C2×C22⋊C8—22no—
88C4×M16—24no—
89C4×M16D414yes—
90C2×C4×C8—22no—
91#400—22no—
92#400—22no—
93#400—24no—
94#398—22no—
95#398—22no—
96#395—24no—
97#391—22no—
98#391—22no—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2×C412yes—
103#389—22no—
104#389—22no—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#385—24no—
110#383—21no—
111#382—21no—
112#382C2×C412yes—
113#382C2×C412yes—
114#381—24no—
115#381—21no—
116#380—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C412yes—
120#376C2312yes—
121Q8⋊C8—216no—
122#324—24no—
123#324—24no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—42no—
129C22×Q8C22⋊C412yes—
130C22×Q8C22×C412yes—
131C22×Q8C22×C411yescharacteristic
132C22×Q8C22×C411yescharacteristic
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C22×C8—42no—
140C4⋊Q8—28no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—44no—
146#125—42no—
147#125—42no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—42no—
171#117—42no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—28no—
215C4⋊C8—28no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—42no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224#97—28no—
225#97—24no—
226C4×C8—28no—
227C4×C8—24no—
228C4×C8—42no—
229C4×C8C2×D418yes—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241C24C2×C22⋊C411yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—44no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8#13114yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C4×D412yes—
282C22×C4C4×D412yes—
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12112yes—
286C22×C4#12112yes—
287C22×C4#12311yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—48no—
293C2×C8—28no—
294C2×C8—44no—
295C2×C8—84no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—48no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—48no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—24no—
327C42—24no—
328C42—44no—
329C42C4×D412yes—
330C42C4×D412yes—
331C42#12012yes—
332C42#12112yes—
333C42#12112yes—
334C42#12312yes—
335C42C22×D412yes—
336C42C2×#4112yes—
337C23—21no—
338C23—21no—
339C23—22no—
340C23—41no—
341C23#38611yescharacteristic
342C23#38611yescharacteristic
343C23C22×C22⋊C411yescharacteristic
344Q8—28no—
345Q8—48no—
346Q8—44no—
347Q8—48no—
348Q8—82no—
349Q8—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4#38611yescharacteristic
389C2×C4#38612yes—
390C2×C4#38612yes—
391C2×C4#38611yescharacteristic
392C2×C4#40814yes—
393C2×C4C2×C4×D412yes—
394C2×C4C2×#12011yescharacteristic
395C2×C4C2×#12112yes—
396C2×C4C2×#12311yescharacteristic
397C2×C4C2×#13112yes—
398C2×C4#56812yes—
399C8—48no—
400C8—84no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—42no—
409C22—42no—
410C22#190912yes—
411C22C2×#38611yescharacteristic
412C22C2×#40811yescharacteristic
413C22#287911yescharacteristic
414C4—22no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—24no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4#191212yes—
429C4#293314yes—
430C2—21no—
431C2—21no—
432C2—22no—
433C2—41no—
434C2#2089711yescharacteristic
435C2C2×#190911yescharacteristic
436C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9104C2semidirect
6. #9100C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#352482
#174642
C2×#14612
#91042
#91002
#78572