Finite Group Databasebeta

← #242371

Subgroups — #242371

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17819C211yesmaximal, characteristic
2#17809C211yesmaximal, characteristic
3C2×#1574C211yesmaximal, characteristic
4#17751C211yesmaximal, characteristic
5#17718C211yesmaximal, characteristic
6#17708C211yesmaximal, characteristic
7#10614C214yesmaximal, complemented
8#10613C214yesmaximal, complemented
9#8129C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1574—22no—
18#1574—22no—
19#1574C2214yes—
20#1574C2214yes—
21#1574C2214yes—
22#1571—21no—
23#1570—21no—
24#1568—22no—
25#1565—22no—
26C2×#335—21no—
27C2×#334—21no—
28#1561—22no—
29#1554—22no—
30#1552—22no—
31#1550—21no—
32#1550—21no—
33#1549C2212yes—
34#1549C2212yes—
35#1548C2212yes—
36#1548C2212yes—
37#1546C2212yes—
38#1546C2212yes—
39#1542—22no—
40#1542—21no—
41#1542—21no—
42#1309C2214yes—
43C22×M16—21no—
44C22×M16—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533Q812yes—
52#533Q812yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×#117—21no—
61C22×C42C2311yescharacteristic
62#432—21no—
63#432—21no—
64#432—21no—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431D412yes—
74#431D412yes—
75#431C2312yes—
76#431C2312yes—
77#431C2312yes—
78#423—21no—
79#423—21no—
80#423—21no—
81#423—21no—
82#423—21no—
83#423—21no—
84#423D412yes—
85#423C2312yes—
86#423C2312yes—
87#423C2312yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96#407—21no—
97#407—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C4×M16—21no—
101C4×M16—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#335—24no—
105#335—24no—
106#335—24no—
107#334—24no—
108#334—24no—
109#334—24no—
110#333—22no—
111#333—22no—
112#333—22no—
113#333—22no—
114#332—22no—
115#332—22no—
116#332—22no—
117#332—22no—
118#332—24no—
119#332—24no—
120C22×Q8—21no—
121C22×Q8—21no—
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—82no—
132C2×M16—82no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—41no—
138C22×C8—41no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8C2×Q812yes—
146C4⋊Q8C2×Q812yes—
147#125—22no—
148#125—22no—
149#125—41no—
150#125—41no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C22⋊Q8—41no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167#117—22no—
168#117—22no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42C2×Q811yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42C2411yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8C2×D414yes—
207C4⋊C8C2×D412yes—
208C4⋊C8C2×D412yes—
209C4⋊C8C2×D412yes—
210C4⋊C8C2×D412yes—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219#97—22no—
220#97—41no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—41no—
227C24#12611yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—82no—
233C2×Q8—81no—
234C2×Q8—81no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4#12611yescharacteristic
256C22×C4#12611yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260M16—44no—
261M16—82no—
262M16—82no—
263M16—82no—
264M16—82no—
265M16—82no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—82no—
279C2×C8—82no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C42—42no—
295C42—42no—
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42C22×D411yescharacteristic
309C42C22×D411yescharacteristic
310C42C22×D411yescharacteristic
311C42C22×Q811yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23#39611yescharacteristic
316C23#39611yescharacteristic
317C23C2×#12611yescharacteristic
318Q8—82no—
319Q8—82no—
320Q8—82no—
321Q8—81no—
322Q8—81no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#39611yescharacteristic
350C2×C4#39611yescharacteristic
351C2×C4#39611yescharacteristic
352C2×C4#39611yescharacteristic
353C2×C4#39611yescharacteristic
354C2×C4#39611yescharacteristic
355C2×C4#47112yes—
356C2×C4#47312yes—
357C2×C4C2×#12111yescharacteristic
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×C22⋊Q811yescharacteristic
361C2×C4C2×C22⋊Q811yescharacteristic
362C2×C4C2×C22⋊Q811yescharacteristic
363C8—42no—
364C8—42no—
365C8—44no—
366C8—42no—
367C8—42no—
368C8—82no—
369C8—82no—
370C8—82no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—82no—
380C22#206412yes—
381C22C2×#39611yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—82no—
392C4#206412yes—
393C4C2×#47111yescharacteristic
394C4C2×#47311yescharacteristic
395C4#306712yes—
396C4#306811yescharacteristic
397C4#306911yescharacteristic
398C2—22no—
399C2—41no—
400C2—41no—
401C2C2×#206411yescharacteristic
402C2#2360111yescharacteristic
403C2#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. #10614C2semidirect
8. #10613C2semidirect

Maximal subgroups

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

subgroupindex
#178192
#178092
C2×#15742
#177512
#177182
#177082
#106142
#106132
#81292