Finite Group Databasebeta

← #120405

Subgroups — #120405

Order 512.

474 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1574C211yesmaximal, characteristic
2C2×#1574C211yesmaximal, characteristic
3C2×#1309C211yesmaximal, characteristic
4#8043C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7880C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#1574—22no—
16#1574—22no—
17#1574—22no—
18#1574C414yescomplemented
19C2×#335—21no—
20C2×#335—21no—
21C2×#334—21no—
22C2×#334—21no—
23#1309—22no—
24#1296—21no—
25#1296—21no—
26#1296—21no—
27#1296—21no—
28#1296—21no—
29#1296—21no—
30#1277—21no—
31#1277—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C42C2311yescharacteristic
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431C2×C412yes—
48#431C2×C412yes—
49#431D412yes—
50#431Q812yes—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423—21no—
60#423C2×C412yes—
61#423D412yes—
62#423Q812yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D411yescharacteristic
70C2×C4⋊C8D411yescharacteristic
71C2×C4⋊C8D411yescharacteristic
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384Q812yes—
85#335—24no—
86#335—24no—
87#334—24no—
88#334—24no—
89C23×C4—41no—
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—81no—
100C2×M16—81no—
101C2×M16—81no—
102C2×M16—81no—
103C2×M16—81no—
104C2×M16—81no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—81no—
122C2×C4⋊C4—81no—
123C2×C4⋊C4—81no—
124C2×C4⋊C4—81no—
125C2×C4⋊C4—81no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×Q811yescharacteristic
133C2×C42C2×Q811yescharacteristic
134C2×C42C2×Q811yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42#4111yescharacteristic
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—22no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8C22⋊C412yes—
159C4⋊C8C22⋊C412yes—
160C4⋊C8C4⋊C412yes—
161C4⋊C8C4⋊C412yes—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—41no—
173C4×C8—41no—
174C24#12611yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4#11711yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4C4×Q811yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4C22⋊Q811yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4#12311yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4#12411yescharacteristic
205C22×C4#12511yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12611yescharacteristic
209M16—82no—
210M16—82no—
211M16—82no—
212M16—82no—
213M16—82no—
214M16—82no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C42C2×C22⋊C411yescharacteristic
252C42C2×C4⋊C411yescharacteristic
253C42C4×D411yescharacteristic
254C42C4×Q811yescharacteristic
255C42#12011yescharacteristic
256C42#12111yescharacteristic
257C42#12111yescharacteristic
258C42#12111yescharacteristic
259C42C22⋊Q811yescharacteristic
260C42C22⋊Q811yescharacteristic
261C42C22⋊Q811yescharacteristic
262C42#12311yescharacteristic
263C42#12511yescharacteristic
264C42C4⋊Q811yescharacteristic
265C42C4⋊Q811yescharacteristic
266C42C4⋊Q811yescharacteristic
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#38711yescharacteristic
274C23#39611yescharacteristic
275C23#40011yescharacteristic
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—164no—
310C2×C4#38011yescharacteristic
311C2×C4#38611yescharacteristic
312C2×C4#38711yescharacteristic
313C2×C4#38911yescharacteristic
314C2×C4#38911yescharacteristic
315C2×C4#39111yescharacteristic
316C2×C4#39411yescharacteristic
317C2×C4#39511yescharacteristic
318C2×C4#39611yescharacteristic
319C2×C4#39811yescharacteristic
320C2×C4#39811yescharacteristic
321C2×C4#39811yescharacteristic
322C2×C4#47111yescharacteristic
323C2×C4#47111yescharacteristic
324C2×C4#47111yescharacteristic
325C2×C4#47111yescharacteristic
326C2×C4#47111yescharacteristic
327C2×C4#47111yescharacteristic
328C2×C4#47211yescharacteristic
329C2×C4#47211yescharacteristic
330C2×C4#47211yescharacteristic
331C2×C4#47311yescharacteristic
332C2×C4#47311yescharacteristic
333C2×C4#47311yescharacteristic
334C8—42no—
335C8—42no—
336C8—42no—
337C8—42no—
338C8—81no—
339C8—82no—
340C8—81no—
341C8—82no—
342C8—82no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22#144111yescharacteristic
354C22#206311yescharacteristic
355C22#206311yescharacteristic
356C22#206311yescharacteristic
357C22#206411yescharacteristic
358C22#206411yescharacteristic
359C22#206411yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—162no—
367C4#181511yescharacteristic
368C4#181611yescharacteristic
369C4#190711yescharacteristic
370C4#190811yescharacteristic
371C4#195111yescharacteristic
372C4#195211yescharacteristic
373C4#205111yescharacteristic
374C4#205211yescharacteristic
375C4#206311yescharacteristic
376C4#206411yescharacteristic
377C4#208911yescharacteristic
378C4#209011yescharacteristic
379C2—41no—
380C2—41no—
381C2#1061211yescharacteristic
382C2#1061311yescharacteristic
383C2#1061411yescharacteristic
384C2#1061411yescharacteristic
385C2#1121611yescharacteristic
386C2#1124211yescharacteristic
387C2#1124911yescharacteristic
subgroupG/H
center, socleC23#396
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
18. #1574C4semidirect

Maximal subgroups

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

subgroupindex
C2×#15742
C2×#13092
#80432
#78802