Finite Group Databasebeta

← #202528

Subgroups — #202528

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425C211yesmaximal, characteristic
2#17787C211yesmaximal, characteristic
3#17464C211yesmaximal, characteristic
4C2×#1485C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic
6#9551C214yesmaximal, complemented
7#9549C214yesmaximal, complemented
8#7936C211yesmaximal, characteristic
9#7887C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1997C412yes—
18#1966C412yes—
19#1901C412yes—
20#1901C412yes—
21C4×#101C2214yes—
22#1565—22no—
23#1565C2212yes—
24#1565C2212yes—
25#1564—22no—
26#1485—22no—
27#1485—21no—
28#1485—21no—
29#1485—22no—
30#1481—22no—
31#1479C2212yes—
32#1479C2212yes—
33C2×Q8⋊C8C2214yes—
34#1453—22no—
35#1451—21no—
36#1451C2212yes—
37C8×M16—21no—
38C8×M16C2212yes—
39#1283C2214yes—
40#1278C2214yes—
41C22×M16—21no—
42C22×M16C2×C411yescharacteristic
43C22×M16C2×C411yescharacteristic
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#432—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#431D412yes—
75#431C2312yes—
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78#423—21no—
79#423C2×C412yes—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8C2312yes—
82C2×#101C2×C414yes—
83C2×#101C2×C414yes—
84#407—21no—
85#407C2×C412yes—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16—21no—
91C4×M16—21no—
92C4×M16—21no—
93C4×M16C2×C412yes—
94C4×M16D411yescharacteristic
95C4×M16D411yescharacteristic
96C4×M16D412yes—
97C4×M16C2312yes—
98C4×M16C2312yes—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105#343C2×C414yes—
106#343C2×C414yes—
107#335—22no—
108#335—22no—
109#334—22no—
110#334—22no—
111Q8⋊C8—24no—
112Q8⋊C8—24no—
113Q8⋊C8—24no—
114Q8⋊C8—24no—
115#322—22no—
116#322—22no—
117#322—22no—
118C82—22no—
119C22×Q8—21no—
120C22×Q8C22×C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—24no—
127C2×M16—21no—
128C2×M16—22no—
129C2×M16—21no—
130C2×M16—42no—
131C2×M16—44no—
132C2×M16—41no—
133C2×M16C22×C412yes—
134C2×M16C22×C414yes—
135C2×M16C22×C412yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149#125—41no—
150#125—41no—
151#125—41no—
152#125—41no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—41no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C22⋊Q8—41no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172#117—41no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22×C412yes—
179C2×C22⋊C4—22no—
180C2×C42—22no—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42C2411yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8C2×D412yes—
201C4⋊C8C2×D412yes—
202#101—28no—
203#101—44no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213#97—22no—
214#97—21no—
215#97—22no—
216#97—21no—
217#97—22no—
218#97—22no—
219#97—41no—
220C4×C8—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×C8—41no—
230C4×C8C2×D412yes—
231C4×C8C2×D412yes—
232C24#11711yescharacteristic
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—82no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
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—24no—
265M16—24no—
266M16—22no—
267M16—24no—
268M16—22no—
269M16—24no—
270M16—22no—
271M16—44no—
272M16—42no—
273M16—42no—
274M16—82no—
275M16#13114yes—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—82no—
289C2×C8—82no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C42—42no—
305C42—42no—
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42#12311yescharacteristic
313C42#12311yescharacteristic
314C42#12411yescharacteristic
315C42#12411yescharacteristic
316C42#12611yescharacteristic
317C42#12611yescharacteristic
318C42C22×D411yescharacteristic
319C42C22×D411yescharacteristic
320C42C2×#4111yescharacteristic
321C42C2×#4111yescharacteristic
322C23—22no—
323C23—41no—
324C23—41no—
325C23#38811yescharacteristic
326C23#38811yescharacteristic
327C23C2×#11711yescharacteristic
328Q8—42no—
329Q8—42no—
330Q8—44no—
331Q8—82no—
332Q8—82no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4C2×C4×D411yescharacteristic
363C2×C4C2×C4×D411yescharacteristic
364C2×C4C2×#12111yescharacteristic
365C2×C4C2×#12311yescharacteristic
366C2×C4C2×#12411yescharacteristic
367C2×C4C2×#12611yescharacteristic
368C2×C4C2×#13112yes—
369C2×C4#56812yes—
370C8—24no—
371C8—24no—
372C8—42no—
373C8—42no—
374C8—44no—
375C8—42no—
376C8—42no—
377C8—82no—
378C8—82no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—82no—
388C22C2×#38811yescharacteristic
389C22#292512yes—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—82no—
400C4#193212yes—
401C4#292911yescharacteristic
402C4#293011yescharacteristic
403C4#293512yes—
404C4#306912yes—
405C2—22no—
406C2—41no—
407C2—41no—
408C2#2105211yescharacteristic
409C2#2107711yescharacteristic
410C2C2×#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
#177872
#174642
C2×#14852
#173272
#95512
#95492
#79362
#78872