Finite Group Databasebeta

← #116557

Subgroups — #116557

Order 512.

629 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21375C211yesmaximal, characteristic
2C2×#1568C211yesmaximal, characteristic, complemented
3C2×#1529C212yesmaximal
4C4×#341C211yesmaximal, characteristic, complemented
5C2×#1291C212yesmaximal, complemented
6C2×#529C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2211yescharacteristic
9C2×C4×M16C2212yes—
10C22×C4×C8C2211yescharacteristic
11#1984C411yescharacteristic
12#1984C411yescharacteristic
13#1954C411yescharacteristic
14#1954C411yescharacteristic
15#1869—21no—
16#1869—21no—
17#1750—21no—
18#1750—21no—
19C2×#341—21no—
20C2×#341—21no—
21#1568—24no—
22C2×#335—21no—
23C2×#334—21no—
24#1529—24no—
25#1529—24no—
26C2×#330—22no—
27C2×#329—22no—
28#1291—24no—
29C22×M16—22no—
30C23×C8D411yescharacteristic
31C23×C8D411yescharacteristic
32#529—22no—
33#529—22no—
34#529—22no—
35#529—22no—
36C2×#126—21no—
37C2×#125D411yescharacteristic
38C2×#124D411yescharacteristic
39C2×#123—21no—
40C2×C22⋊Q8Q811yescharacteristic
41C2×#121Q811yescharacteristic
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×D4—21no—
44C2×C4×D4C2×C411yescharacteristic
45C2×#117—21no—
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432—22no—
51#432—22no—
52#432D414yes—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2×C411yescharacteristic
55C2×C4⋊C8C2×C411yescharacteristic
56C2×C22⋊C8—22no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C4×M16—22no—
63C4×M16—22no—
64C4×M16—22no—
65C4×M16D414yes—
66C2×#97—22no—
67C2×#97—22no—
68C2×#97—21no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8C2×C411yescharacteristic
73C2×C4×C8C2×C411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76#341—24no—
77#341—24no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#335—24no—
83#334—24no—
84#330—28no—
85#329—28no—
86C22×Q8C4⋊C411yescharacteristic
87C22×D4—21no—
88C22×D4C4⋊C411yescharacteristic
89C23×C4—21no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×Q811yescharacteristic
92C23×C4#4111yescharacteristic
93C2×M16—24no—
94C2×M16—24no—
95C2×M16—24no—
96C2×M16—42no—
97C2×M16—42no—
98C2×M16—42no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8C22⋊C411yescharacteristic
110C22×C8C22⋊C411yescharacteristic
111C22×C8C22⋊C411yescharacteristic
112C22×C8C22⋊C411yescharacteristic
113#126—42no—
114#126—42no—
115#125—22no—
116#125—22no—
117#124—22no—
118#124—22no—
119#123—42no—
120#123—42no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123#121—22no—
124#121—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—42no—
130C4×D4—42no—
131#117—41no—
132#117—41no—
133#117—41no—
134#117—41no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C4⋊C411yescharacteristic
140C2×C4⋊C4C4⋊C411yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C42—21no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42#4112yes—
153C2×C42#4112yes—
154C2×C42#4111yescharacteristic
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—24no—
158C4⋊C8—42no—
159C22⋊C8—24no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163C22⋊C8—42no—
164C22⋊C8—42no—
165C22⋊C8—42no—
166#97—24no—
167#97—24no—
168#97—42no—
169#97—42no—
170#97—41no—
171#97—41no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—21no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—24no—
179C4×C8—42no—
180C24—21no—
181C24C22⋊Q811yescharacteristic
182C2×Q8—22no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×D4—22no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C4⋊C411yescharacteristic
210C22×C4#11711yescharacteristic
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12112yes—
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q812yes—
218C22×C4C22⋊Q812yes—
219C22×C4#12612yes—
220M16—28no—
221M16—44no—
222M16—44no—
223M16—44no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—44no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8#10412yes—
249C2×C8#10412yes—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C42—41no—
266C42—41no—
267C42C4×D412yes—
268C42C4×D412yes—
269C42#12012yes—
270C42#12011yescharacteristic
271C42#12312yes—
272C42#12311yescharacteristic
273C42#12312yes—
274C42#12412yes—
275C42#12411yescharacteristic
276C42#12711yescharacteristic
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#38011yescharacteristic
291C23#39612yes—
292Q8—82no—
293D4—82no—
294D4—84no—
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—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4#38512yes—
327C2×C4#38611yescharacteristic
328C2×C4#38812yes—
329C2×C4#39011yescharacteristic
330C2×C4#39211yescharacteristic
331C2×C4#39311yescharacteristic
332C2×C4#39412yes—
333C2×C4#39711yescharacteristic
334C2×C4#39911yescharacteristic
335C2×C4C2×#10411yescharacteristic
336C2×C4C2×#10411yescharacteristic
337C2×C4#42111yescharacteristic
338C2×C4#42111yescharacteristic
339C8—22no—
340C8—22no—
341C8—48no—
342C8—41no—
343C8—41no—
344C8—84no—
345C8—81no—
346C8—81no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#143911yescharacteristic
362C22#180611yescharacteristic
363C22#180611yescharacteristic
364C22#181311yescharacteristic
365C22#181311yescharacteristic
366C22#181411yescharacteristic
367C22#181411yescharacteristic
368C4—22no—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—21no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4#187111yescharacteristic
379C4#187111yescharacteristic
380C4#196211yescharacteristic
381C4#196311yescharacteristic
382C2—42no—
383C2—161no—
384C2#869011yescharacteristic
385C2#869111yescharacteristic
386C2#869211yescharacteristic
387C2#869311yescharacteristic
388C2#1105311yescharacteristic
389C2#1109911yescharacteristic
390C2#1110411yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121Q8
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
2. C2×#1568C2semidirect
4. C4×#341C2semidirect
5. C2×#1291C2semidirect

Maximal subgroups

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

subgroupindex
#213752
C2×#15682
C2×#15292
C4×#3412
C2×#12912