Finite Group Databasebeta

← #202516

Subgroups — #202516

Order 512.

689 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20727C211yesmaximal, characteristic
2#17808C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4C2×#1479C211yesmaximal, characteristic
5#17324C211yesmaximal, characteristic
6#9551C214yesmaximal, complemented
7#9549C214yesmaximal, complemented
8#7936C211yesmaximal, characteristic
9#7886C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#1821C412yes—
18#1801C412yes—
19#1801C412yes—
20#1784C412yes—
21C4×#101C2214yes—
22#1574—22no—
23#1572—22no—
24#1565C2214yes—
25#1491—22no—
26#1487—22no—
27#1479—22no—
28#1479—22no—
29#1479C2214yes—
30C2×Q8⋊C8C2212yes—
31C2×Q8⋊C8C2212yes—
32#1454—22no—
33#1453—22no—
34#1451C2212yes—
35C8×M16C2212yes—
36#1283C2214yes—
37#1278C2212yes—
38#1278C2212yes—
39C23×Q8C2×C411yescharacteristic
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432D412yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431D412yes—
77#431D412yes—
78#431C2312yes—
79#431C2312yes—
80#423—21no—
81#423C2×C411yescharacteristic
82#423C2×C411yescharacteristic
83#423C2×C412yes—
84#423D412yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×#101C2×C414yes—
88C2×#101C2×C414yes—
89#407—21no—
90#407—21no—
91#407C2×C412yes—
92#407C2×C412yes—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16C2312yes—
98C4×M16C2312yes—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#343C2×C414yes—
103#343C2×C414yes—
104#335—24no—
105#334—24no—
106Q8⋊C8—28no—
107Q8⋊C8—24no—
108Q8⋊C8—24no—
109#322—22no—
110#322—24no—
111C82—22no—
112C22×Q8—21no—
113C22×Q8—24no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—42no—
119C22×Q8C22×C412yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C22×C411yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—22no—
126C2×M16—41no—
127C2×M16C22×C412yes—
128C2×M16C22×C414yes—
129C2×M16C22×C412yes—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C22×C412yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42C2411yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—41no—
192C4⋊C8C2×D414yes—
193#101—28no—
194#101—44no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207#97—22no—
208#97—22no—
209#97—22no—
210#97—41no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—41no—
218C4×C8—41no—
219C4×C8C2×D412yes—
220C4×C8C2×D412yes—
221C24#11711yescharacteristic
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8#13114yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—81no—
248C22×C4—82no—
249C22×C4#11711yescharacteristic
250C22×C4#11711yescharacteristic
251C22×C4#11711yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4C23×C411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267M16—24no—
268M16—24no—
269M16—24no—
270M16—24no—
271M16—82no—
272M16—82no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—81no—
282C2×C8—81no—
283C2×C8—82no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12311yescharacteristic
303C42#12311yescharacteristic
304C42#12411yescharacteristic
305C42#12411yescharacteristic
306C42#12611yescharacteristic
307C42#12611yescharacteristic
308C42C22×D411yescharacteristic
309C42C22×D411yescharacteristic
310C42C2×#4111yescharacteristic
311C42C2×#4111yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23#38811yescharacteristic
316C23#38811yescharacteristic
317C23C2×#11711yescharacteristic
318Q8—24no—
319Q8—24no—
320Q8—44no—
321Q8—44no—
322Q8—42no—
323Q8—42no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#38811yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C2×C4×D411yescharacteristic
360C2×C4C2×#12111yescharacteristic
361C2×C4C2×#12311yescharacteristic
362C2×C4C2×#12411yescharacteristic
363C2×C4C2×#12611yescharacteristic
364C2×C4C2×#13112yes—
365C2×C4#56812yes—
366C8—42no—
367C8—42no—
368C8—42no—
369C8—42no—
370C8—44no—
371C8—44no—
372C8—82no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—44no—
383C22C2×#38811yescharacteristic
384C22#292412yes—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—44no—
395C4—82no—
396C4#292911yescharacteristic
397C4#293011yescharacteristic
398C4#293312yes—
399C4#306912yes—
400C2—22no—
401C2—41no—
402C2—41no—
403C2#2105011yescharacteristic
404C2#2107711yescharacteristic
405C2#2108011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9551C2semidirect
7. #9549C2semidirect

Maximal subgroups

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

subgroupindex
#207272
#178082
#174492
C2×#14792
#173242
#95512
#95492
#79362
#78862