Finite Group Databasebeta

← #242385

Subgroups — #242385

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17823C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17749C211yesmaximal, characteristic
5#17747C211yesmaximal, characteristic
6#17715C211yesmaximal, characteristic
7#10613C214yesmaximal, complemented
8#10613C214yesmaximal, complemented
9#8128C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1574—22no—
18#1574C2214yes—
19#1574C2214yes—
20#1574C2214yes—
21#1572—22no—
22#1571—21no—
23#1570—21no—
24#1569—22no—
25#1568—22no—
26#1565—22no—
27#1559—22no—
28#1559—22no—
29#1554—22no—
30#1552—21no—
31#1552—21no—
32#1550—22no—
33#1549C2212yes—
34#1549C2212yes—
35#1548C2212yes—
36#1548C2212yes—
37#1546C2212yes—
38#1546C2212yes—
39#1544—22no—
40#1309C2214yes—
41C22×M16—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533Q812yes—
49#533Q812yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×#117—21no—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432D412yes—
66#432D412yes—
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#431D412yes—
76#431C2312yes—
77#431C2312yes—
78#431C2312yes—
79#423—21no—
80#423—21no—
81#423—21no—
82#423D412yes—
83#423D412yes—
84#423C2312yes—
85#423C2312yes—
86#423C2312yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91#407—21no—
92#407—21no—
93#407—21no—
94#407—21no—
95C2×C22⋊C8—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C4×M16—21no—
100C2×C4×C8—21no—
101#335—24no—
102#335—24no—
103#335—24no—
104#334—24no—
105#334—24no—
106#334—24no—
107#333—24no—
108#333—24no—
109#332—22no—
110#332—22no—
111#332—22no—
112#332—22no—
113#332—24no—
114#332—24no—
115C22×Q8—21no—
116C22×Q8—21no—
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—41no—
126C2×M16—82no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—41no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8C2×Q812yes—
137C4⋊Q8C2×Q812yes—
138#125—22no—
139#125—22no—
140#125—41no—
141#125—41no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—22no—
160#117—41no—
161#117—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42—41no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q811yescharacteristic
177C2×C42C2×Q811yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42C2411yescharacteristic
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8C2×D412yes—
195C4⋊C8C2×D412yes—
196C4⋊C8C2×D414yes—
197C4⋊C8C2×D414yes—
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—
207C22⋊C8—41no—
208#97—22no—
209#97—22no—
210#97—41no—
211#97—41no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—41no—
217C4×C8—41no—
218C24#12611yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—82no—
224C2×Q8—81no—
225C2×Q8—81no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12611yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4#12611yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4111yescharacteristic
250M16—44no—
251M16—82no—
252M16—82no—
253M16—82no—
254M16—82no—
255M16—82no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—81no—
264C2×C8—81no—
265C2×C8—82no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—42no—
281C42—42no—
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42C22⋊Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22×D411yescharacteristic
295C42C22×D411yescharacteristic
296C42C22×D411yescharacteristic
297C42C22×Q811yescharacteristic
298C23—41no—
299C23—41no—
300C23—41no—
301C23#39611yescharacteristic
302C23#39611yescharacteristic
303C23C2×#12611yescharacteristic
304Q8—82no—
305Q8—82no—
306Q8—82no—
307Q8—81no—
308Q8—81no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4#39611yescharacteristic
335C2×C4#39611yescharacteristic
336C2×C4#39611yescharacteristic
337C2×C4#39611yescharacteristic
338C2×C4#39611yescharacteristic
339C2×C4#39611yescharacteristic
340C2×C4C2×#12111yescharacteristic
341C2×C4C2×#12111yescharacteristic
342C2×C4C2×#12111yescharacteristic
343C2×C4C2×C22⋊Q811yescharacteristic
344C2×C4C2×C22⋊Q811yescharacteristic
345C2×C4C2×C22⋊Q811yescharacteristic
346C8—44no—
347C8—42no—
348C8—42no—
349C8—44no—
350C8—82no—
351C8—82no—
352C8—82no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—82no—
363C22C2×#39611yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—82no—
375C4#306712yes—
376C4#306811yescharacteristic
377C4#306912yes—
378C4#306911yescharacteristic
379C2—41no—
380C2—41no—
381C2—41no—
382C2#2360111yescharacteristic
383C2#2360311yescharacteristic
384C2#2360411yescharacteristic
subgroupG/H
center, socleC22C2×#396
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ 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
7. #10613C2semidirect
8. #10613C2semidirect

Maximal subgroups

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

subgroupindex
#178232
#178192
#178092
#177492
#177472
#177152
#106132
#81282