Finite Group Databasebeta

← #117368

Subgroups — #117368

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21829C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3#17807C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#8098C211yesmaximal, characteristic
6#8082C211yesmaximal, characteristic
7#7950C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1997—22no—
16#1997—21no—
17#1997—21no—
18#1987—22no—
19#1987—22no—
20#1821—22no—
21C2×#340—22no—
22#1573—22no—
23#1571—21no—
24#1570—21no—
25#1565—22no—
26#1556C414yescomplemented
27#1549—22no—
28#1546—22no—
29#1303—22no—
30#1301—22no—
31#1285—22no—
32C23×Q8—21no—
33C23×C8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8Q812yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D412yes—
49C22×C42—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432C2×C412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431D412yes—
67#423—21no—
68#423D412yes—
69#423D411yescharacteristic
70#423D411yescharacteristic
71#423D412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D412yes—
78#407—21no—
79#407—21no—
80#407—21no—
81#407—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4D412yes—
90C2×#95—21no—
91#343—24no—
92#343—24no—
93#340—28no—
94#335—24no—
95#334—24no—
96#332—28no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—84no—
104C23×C4—22no—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—22no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—22no—
113C2×M16—41no—
114C22×C8—22no—
115C22×C8—44no—
116C22×C8—41no—
117C22×C8—41no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4C4⋊C414yes—
150C2×C42—22no—
151C2×C42—41no—
152C2×C42—82no—
153C2×C42C22×C411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×Q811yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—21no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8C22⋊C414yes—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—41no—
193C4×C8—41no—
194C4×C8—41no—
195#95—22no—
196#95—41no—
197C24#12311yescharacteristic
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—84no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×Q8—84no—
206C2×Q8—84no—
207C2×Q8—84no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—84no—
221C22×C4C2×C4⋊C411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239M16—42no—
240M16—42no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—42no—
245M16—82no—
246C2×C8—24no—
247C2×C8—44no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C42—82no—
267C42—82no—
268C42C2×C22⋊C411yescharacteristic
269C42#11711yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42#12011yescharacteristic
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42C22⋊Q811yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42#12311yescharacteristic
282C42#12411yescharacteristic
283C42#12711yescharacteristic
284C23—22no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23#38511yescharacteristic
289C23#39411yescharacteristic
290C23#39711yescharacteristic
291Q8—84no—
292Q8—84no—
293Q8—84no—
294Q8—84no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—88no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4#38011yescharacteristic
323C2×C4#38511yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#39011yescharacteristic
328C2×C4#39211yescharacteristic
329C2×C4#39311yescharacteristic
330C2×C4#39411yescharacteristic
331C2×C4#39611yescharacteristic
332C2×C4#39611yescharacteristic
333C2×C4#39911yescharacteristic
334C2×C4#45412yes—
335C2×C4#45612yes—
336C2×C4#47112yes—
337C2×C4#47312yes—
338C8—44no—
339C8—82no—
340C8—82no—
341C8—81no—
342C8—81no—
343C8—82no—
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22#143911yescharacteristic
353C22#185712yes—
354C22#204611yescharacteristic
355C22#204711yescharacteristic
356C22#208112yes—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—84no—
366C4—84no—
367C4#190812yes—
368C4#200311yescharacteristic
369C4#200511yescharacteristic
370C4#203812yes—
371C4#206412yes—
372C2—21no—
373C2—22no—
374C2—22no—
375C2—21no—
376C2#1106211yescharacteristic
377C2#1113311yescharacteristic
378C2#1113611yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#218292
C2×#19872
#178072
#177422
#80982
#80822
#79502