Finite Group Databasebeta

← #117398

Subgroups — #117398

Order 512.

558 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
1#17806C211yesmaximal, characteristic
2#17718C211yesmaximal, characteristic
3#17648C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#8084C211yesmaximal, characteristic
6#7935C211yesmaximal, characteristic
7#7873C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1574—22no—
16#1573—22no—
17#1564—22no—
18#1554—22no—
19#1548C411yescharacteristic
20#1548C411yescharacteristic
21#1546C411yescharacteristic
22#1546C411yescharacteristic
23#1542—22no—
24#1531—22no—
25#1528—22no—
26#1526—22no—
27#1481—22no—
28#1481—22no—
29C2×Q8⋊C8—22no—
30#1302—22no—
31#1283—22no—
32#1277—22no—
33C22×M16—21no—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533D412yes—
42#533Q812yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432D412yes—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C411yescharacteristic
66#431C2×C411yescharacteristic
67#431D412yes—
68#431D412yes—
69#423—21no—
70#423C2×C411yescharacteristic
71#423C2×C411yescharacteristic
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D412yes—
79C2×C4⋊C8D412yes—
80#407—21no—
81#407—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C2×#97D412yes—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#335—24no—
95#334—24no—
96#333—22no—
97#333—22no—
98#332—22no—
99#332—22no—
100#330—28no—
101Q8⋊C8—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—24no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—42no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—82no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4⋊C412yes—
128C4⋊Q8C4⋊C412yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42—41no—
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8C22⋊C412yes—
187C4⋊C8C22⋊C412yes—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—41no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—41no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210C24#12611yescharacteristic
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—81no—
225C22×C4#11711yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12611yescharacteristic
243M16—44no—
244M16—44no—
245M16—42no—
246M16—42no—
247M16—82no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—42no—
279C42—42no—
280C42C2×C22⋊C411yescharacteristic
281C42C2×C4⋊C411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42#12011yescharacteristic
285C42#12011yescharacteristic
286C42#12011yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42C22⋊Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42#12311yescharacteristic
294C42#12411yescharacteristic
295C42#12711yescharacteristic
296C23—22no—
297C23—22no—
298C23—41no—
299C23#38811yescharacteristic
300C23#39611yescharacteristic
301C23#39711yescharacteristic
302Q8—44no—
303Q8—82no—
304Q8—82no—
305Q8—81no—
306Q8—81no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4#38011yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#39011yescharacteristic
337C2×C4#39211yescharacteristic
338C2×C4#39311yescharacteristic
339C2×C4#39411yescharacteristic
340C2×C4#39411yescharacteristic
341C2×C4#39611yescharacteristic
342C2×C4#39911yescharacteristic
343C2×C4#44312yes—
344C2×C4#44412yes—
345C2×C4#47112yes—
346C2×C4#47312yes—
347C8—44no—
348C8—42no—
349C8—42no—
350C8—82no—
351C8—82no—
352C8—81no—
353C8—81no—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—44no—
363C22#143911yescharacteristic
364C22#192912yes—
365C22#206412yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—82no—
376C4#186612yes—
377C4#190812yes—
378C4#197411yescharacteristic
379C4#197511yescharacteristic
380C4#204611yescharacteristic
381C4#204711yescharacteristic
382C2—22no—
383C2—22no—
384C2—41no—
385C2#1106211yescharacteristic
386C2#1112511yescharacteristic
387C2#1114111yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#178062
#177182
#176482
C2×#14812
#80842
#79352
#78732