Finite Group Databasebeta

← #202504

Subgroups — #202504

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425C211yesmaximal, characteristic
2#17808C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4C2×#1485C211yesmaximal, characteristic
5M162C211yesmaximal, characteristic
6#9551C214yesmaximal, complemented
7#9549C214yesmaximal, complemented
8#7932C211yesmaximal, characteristic
9#7883C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1997C412yes—
18#1966C412yes—
19#1901C414yes—
20C4×#101C2212yes—
21C4×#101C2212yes—
22#1574—22no—
23#1572—22no—
24#1565C2214yes—
25#1497—22no—
26#1493—22no—
27#1485—21no—
28#1485—21no—
29#1485—22no—
30#1479C2214yes—
31C2×Q8⋊C8C2214yes—
32#1455—22no—
33#1451—21no—
34#1451C2212yes—
35C8×M16—21no—
36C8×M16C2212yes—
37#1283C2212yes—
38#1283C2212yes—
39#1278C2214yes—
40C22×M16—21no—
41C22×M16C2×C412yes—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8C2312yes—
57C2×#117—21no—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432D412yes—
63#431—21no—
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#431C2312yes—
74#431C2312yes—
75#423C2×C412yes—
76#423D412yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2312yes—
79C2×#101C2×C414yes—
80C2×#101C2×C414yes—
81#407—21no—
82#407C2×C412yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16C2×C412yes—
91C4×M16D411yescharacteristic
92C4×M16D411yescharacteristic
93C4×M16D412yes—
94C4×M16C2311yescharacteristic
95C4×M16C2311yescharacteristic
96C4×M16C2312yes—
97C2×#97—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103#343C2×C414yes—
104#343C2×C414yes—
105#335—24no—
106#334—24no—
107Q8⋊C8—28no—
108Q8⋊C8—24no—
109Q8⋊C8—24no—
110#322—22no—
111#322—22no—
112#322—22no—
113C82—22no—
114C22×Q8—21no—
115C22×Q8C22×C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—21no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—21no—
123C2×M16—22no—
124C2×M16—24no—
125C2×M16—42no—
126C2×M16—44no—
127C2×M16C22×C412yes—
128C2×M16C22×C414yes—
129C2×M16C22×C412yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142#125—41no—
143#125—41no—
144#125—41no—
145#125—41no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
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—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—22no—
173C2×C42—22no—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—41no—
193C4⋊C8C2×D414yes—
194#101—28no—
195#101—44no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—22no—
205#97—22no—
206#97—21no—
207#97—22no—
208#97—21no—
209#97—21no—
210#97—21no—
211#97—22no—
212C4×C8—22no—
213C4×C8—21no—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—41no—
220C4×C8—41no—
221C4×C8—41no—
222C4×C8C2×D412yes—
223C4×C8C2×D412yes—
224C24#11711yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4#11711yescharacteristic
239C22×C4#11711yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4C23×C411yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256M16—24no—
257M16—22no—
258M16—24no—
259M16—22no—
260M16—22no—
261M16—24no—
262M16—44no—
263M16—44no—
264M16—42no—
265M16—82no—
266M16#13114yes—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—82no—
278C2×C8—82no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C42—42no—
294C42—42no—
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12311yescharacteristic
302C42#12311yescharacteristic
303C42#12411yescharacteristic
304C42#12411yescharacteristic
305C42#12611yescharacteristic
306C42#12611yescharacteristic
307C42C22×D411yescharacteristic
308C42C22×D411yescharacteristic
309C42C2×#4111yescharacteristic
310C42C2×#4111yescharacteristic
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23#38811yescharacteristic
316C23#38811yescharacteristic
317C23C2×#11711yescharacteristic
318Q8—44no—
319Q8—44no—
320Q8—82no—
321Q8—82no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4C2×C4×D411yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×#12111yescharacteristic
354C2×C4C2×#12311yescharacteristic
355C2×C4C2×#12411yescharacteristic
356C2×C4C2×#12611yescharacteristic
357C2×C4C2×#13112yes—
358C2×C4#56812yes—
359C8—24no—
360C8—24no—
361C8—42no—
362C8—42no—
363C8—44no—
364C8—44no—
365C8—82no—
366C8—82no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—81no—
376C22—81no—
377C22C2×#38811yescharacteristic
378C22#292211yescharacteristic
379C22#292311yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—82no—
390C4#193212yes—
391C4#293312yes—
392C4#293512yes—
393C4#306912yes—
394C2—21no—
395C2—21no—
396C2—41no—
397C2—41no—
398C2#2104811yescharacteristic
399C2#2107711yescharacteristic
400C2C2×#193211yescharacteristic
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
#214252
#178082
#174792
C2×#14852
M1622
#95512
#95492
#79322
#78832