Finite Group Databasebeta

← #202500

Subgroups — #202500

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425C211yesmaximal, characteristic
2#17807C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic
6#9551C214yesmaximal, complemented
7#9549C214yesmaximal, complemented
8C2×#1283C211yesmaximal, characteristic
9#7884C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1997C412yes—
18#1966C412yes—
19#1901C414yes—
20C4×#101C2212yes—
21C4×#101C2212yes—
22#1573—22no—
23#1571—21no—
24#1570—21no—
25#1565C2214yes—
26#1497—22no—
27#1493—21no—
28#1493—21no—
29#1481—21no—
30#1481—21no—
31#1481—22no—
32#1479C2214yes—
33C2×Q8⋊C8C2214yes—
34#1453—22no—
35#1451—21no—
36#1451C2212yes—
37C8×M16—21no—
38C8×M16C2212yes—
39#1283C2212yes—
40#1283C2212yes—
41#1278C2214yes—
42C22×M16C2×C412yes—
43C23×C8—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8C2312yes—
59C2×#117—21no—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432D412yes—
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#431D411yescharacteristic
74#431D411yescharacteristic
75#431D412yes—
76#431C2312yes—
77#431C2312yes—
78#423C2×C412yes—
79#423D411yescharacteristic
80#423D411yescharacteristic
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×#101C2×C414yes—
90C2×#101C2×C414yes—
91#407—21no—
92#407C2×C412yes—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16C2×C412yes—
99C4×M16C2311yescharacteristic
100C4×M16C2311yescharacteristic
101C4×M16C2312yes—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2312yes—
106#343C2×C414yes—
107#343C2×C414yes—
108#335—24no—
109#334—24no—
110Q8⋊C8—24no—
111Q8⋊C8—24no—
112Q8⋊C8—24no—
113Q8⋊C8—24no—
114#322—22no—
115#322—22no—
116#322—22no—
117C82—22no—
118C22×Q8—21no—
119C22×Q8C22×C412yes—
120C23×C4C22×C411yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C2×M16—22no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—24no—
127C2×M16—42no—
128C2×M16C22×C412yes—
129C2×M16C22×C414yes—
130C2×M16C22×C412yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—82no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146#125—41no—
147#125—41no—
148#125—41no—
149#125—41no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22×C412yes—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42C2411yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8C2×D412yes—
204C4⋊C8C2×D412yes—
205#101—28no—
206#101—44no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216#97—22no—
217#97—22no—
218#97—21no—
219#97—21no—
220#97—22no—
221C4×C8—22no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8—21no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—41no—
229C4×C8C2×D412yes—
230C4×C8C2×D412yes—
231C24#11711yescharacteristic
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—82no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—81no—
246C22×C4#11711yescharacteristic
247C22×C4#11711yescharacteristic
248C22×C4#11711yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4C23×C411yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264M16—22no—
265M16—24no—
266M16—22no—
267M16—22no—
268M16—24no—
269M16—44no—
270M16—42no—
271M16—42no—
272M16#13114yes—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—44no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—82no—
287C2×C8—82no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C42—42no—
303C42—42no—
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12311yescharacteristic
311C42#12311yescharacteristic
312C42#12411yescharacteristic
313C42#12411yescharacteristic
314C42#12611yescharacteristic
315C42#12611yescharacteristic
316C42C22×D411yescharacteristic
317C42C22×D411yescharacteristic
318C42C2×#4111yescharacteristic
319C42C2×#4111yescharacteristic
320C23—21no—
321C23—21no—
322C23—22no—
323C23—41no—
324C23#38811yescharacteristic
325C23#38811yescharacteristic
326C23C2×#11711yescharacteristic
327Q8—44no—
328Q8—44no—
329Q8—82no—
330Q8—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#47112yes—
362C2×C4#47312yes—
363C2×C4C2×C4×D411yescharacteristic
364C2×C4C2×C4×D411yescharacteristic
365C2×C4C2×#12111yescharacteristic
366C2×C4C2×#12311yescharacteristic
367C2×C4C2×#12411yescharacteristic
368C2×C4C2×#12611yescharacteristic
369C2×C4C2×#13112yes—
370C2×C4#56812yes—
371C8—24no—
372C8—42no—
373C8—42no—
374C8—42no—
375C8—44no—
376C8—42no—
377C8—42no—
378C8—82no—
379C8—82no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22#193012yes—
390C22C2×#38811yescharacteristic
391C22#292211yescharacteristic
392C22#292311yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—82no—
402C4—82no—
403C4#193012yes—
404C4#293312yes—
405C4#293512yes—
406C4C2×#47111yescharacteristic
407C4C2×#47311yescharacteristic
408C2—21no—
409C2—21no—
410C2—22no—
411C2—41no—
412C2#2104811yescharacteristic
413C2C2×#193011yescharacteristic
414C2#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
#214252
#178072
#174792
C2×#14812
#173272
#95512
#95492
C2×#12832
#78842