Finite Group Databasebeta

← #242380

Subgroups — #242380

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17820C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#17808C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#17742C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10614C214yesmaximal, complemented
8#10613C214yesmaximal, complemented
9#8128C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1574—22no—
18#1574C2212yes—
19#1574C2214yes—
20#1574C2214yes—
21#1574C2212yes—
22#1572—22no—
23#1571—21no—
24#1570—21no—
25#1569—22no—
26#1568—22no—
27#1565—21no—
28#1565—21no—
29#1556—22no—
30#1556—22no—
31#1549—21no—
32#1549—21no—
33#1549C2212yes—
34#1549C2212yes—
35#1548—22no—
36#1548C2214yes—
37#1546—22no—
38#1546C2214yes—
39C2×#333—21no—
40C2×#333—21no—
41#1309C2212yes—
42#1309C2212yes—
43C23×Q8—21no—
44C22×M16—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8Q812yes—
55C2×C4⋊Q8Q812yes—
56C2×C4⋊Q8C2312yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C42C2311yescharacteristic
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432D412yes—
67#432D412yes—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431D412yes—
77#431C2312yes—
78#431C2312yes—
79#431C2312yes—
80#423—21no—
81#423—21no—
82#423—21no—
83#423D412yes—
84#423D411yescharacteristic
85#423D411yescharacteristic
86#423C2312yes—
87#423C2312yes—
88#423C2311yescharacteristic
89#423C2311yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94#407—21no—
95#407—21no—
96#407—21no—
97#407—21no—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C2×C4×C8—21no—
104#335—22no—
105#335—22no—
106#335—24no—
107#335—24no—
108#334—22no—
109#334—22no—
110#334—24no—
111#334—24no—
112#333—24no—
113#333—24no—
114#332—24no—
115#332—24no—
116#332—24no—
117#332—24no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—41no—
124C22×Q8—84no—
125C23×C4—41no—
126C23×C4#4111yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—21no—
130C2×M16—21no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—41no—
137C2×M16—82no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—41no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8—41no—
156C4⋊Q8C2×Q814yes—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q811yescharacteristic
176C2×C42C2×Q811yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42C2411yescharacteristic
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8C2×D412yes—
197C4⋊C8C2×D414yes—
198C4⋊C8C2×D412yes—
199C4⋊C8C2×D412yes—
200C4⋊C8C2×D412yes—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211#97—22no—
212#97—22no—
213#97—41no—
214#97—41no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—41no—
220C4×C8—41no—
221C24#12611yescharacteristic
222C2×Q8—24no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×Q8—84no—
229C2×Q8—84no—
230C2×Q8—84no—
231C2×Q8—84no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—84no—
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4#12611yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259M16—42no—
260M16—42no—
261M16—82no—
262M16—82no—
263M16—82no—
264M16—81no—
265M16—81no—
266M16—82no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—82no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42#12111yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22×D411yescharacteristic
300C42C22×D411yescharacteristic
301C42C22×D411yescharacteristic
302C42C22×Q811yescharacteristic
303C23—22no—
304C23—41no—
305C23—41no—
306C23#39611yescharacteristic
307C23#39611yescharacteristic
308C23C2×#12611yescharacteristic
309Q8—84no—
310Q8—84no—
311Q8—84no—
312Q8—84no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4#39611yescharacteristic
340C2×C4#39611yescharacteristic
341C2×C4#39611yescharacteristic
342C2×C4#39611yescharacteristic
343C2×C4#39611yescharacteristic
344C2×C4#39611yescharacteristic
345C2×C4#47112yes—
346C2×C4#47312yes—
347C2×C4C2×#12111yescharacteristic
348C2×C4C2×#12111yescharacteristic
349C2×C4C2×#12111yescharacteristic
350C2×C4C2×C22⋊Q811yescharacteristic
351C2×C4C2×C22⋊Q811yescharacteristic
352C2×C4C2×C22⋊Q811yescharacteristic
353C8—42no—
354C8—42no—
355C8—44no—
356C8—42no—
357C8—42no—
358C8—82no—
359C8—82no—
360C8—81no—
361C8—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#206412yes—
372C22C2×#39611yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—84no—
382C4#206412yes—
383C4C2×#47111yescharacteristic
384C4C2×#47311yescharacteristic
385C4#306812yes—
386C4#306911yescharacteristic
387C4#306911yescharacteristic
388C2—22no—
389C2—41no—
390C2—41no—
391C2C2×#206411yescharacteristic
392C2#2360211yescharacteristic
393C2#2360311yescharacteristic
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
#178202
#178192
#178082
#177422
C2×#15482
#106142
#106132
#81282