Finite Group Databasebeta

← #117258

Subgroups — #117258

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17817C211yesmaximal, characteristic
2#17742C211yesmaximal, characteristic
3C2×#1548C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5#8004C211yesmaximal, characteristic
6#7934C211yesmaximal, characteristic
7#7884C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#1574—22no—
16#1573—22no—
17#1567—21no—
18#1566—21no—
19#1556—22no—
20#1549—22no—
21#1548—22no—
22#1548—22no—
23#1546—22no—
24#1546—21no—
25#1546—21no—
26#1546—22no—
27C2×#333C414yes—
28C2×#332—22no—
29#1292—22no—
30#1283—22no—
31#1278—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C22×M16—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⋊Q8D412yes—
46C2×C4⋊Q8Q812yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431D412yes—
64#431D412yes—
65#431D412yes—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74#423D411yescharacteristic
75#423D411yescharacteristic
76#423D412yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2×C412yes—
81C2×C4⋊C8C2×C412yes—
82C2×C4⋊C8D412yes—
83#407—21no—
84#407—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C4×M16—21no—
88C4×M16—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#335—24no—
92#334—24no—
93#333—28no—
94#332—28no—
95#332—24no—
96#332—24no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—82no—
104C22×Q8—82no—
105C23×C4—41no—
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—22no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—82no—
117C2×M16—82no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—41no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8C4⋊C414yes—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C42C22×C411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×Q811yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—22no—
170C4⋊C8—21no—
171C4⋊C8—22no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8C22⋊C414yes—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188#97—41no—
189#97—41no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8—41no—
196C24#12611yescharacteristic
197C2×Q8—24no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—84no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×Q8—84no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—84no—
208C2×Q8—82no—
209C2×Q8—82no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—84no—
219C22×C4#11711yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12611yescharacteristic
237M16—42no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—82no—
242M16—82no—
243M16—82no—
244M16—82no—
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—82no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C42C2×C22⋊C411yescharacteristic
265C42C2×C4⋊C411yescharacteristic
266C42C4×D411yescharacteristic
267C42C4×D411yescharacteristic
268C42#12011yescharacteristic
269C42#12011yescharacteristic
270C42#12011yescharacteristic
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42#12111yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42#12311yescharacteristic
278C42#12411yescharacteristic
279C42#12711yescharacteristic
280C23—22no—
281C23—22no—
282C23—41no—
283C23#38811yescharacteristic
284C23#39611yescharacteristic
285C23#39711yescharacteristic
286Q8—84no—
287Q8—84no—
288Q8—84no—
289Q8—82no—
290Q8—82no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4#38011yescharacteristic
316C2×C4#38511yescharacteristic
317C2×C4#38511yescharacteristic
318C2×C4#38611yescharacteristic
319C2×C4#38811yescharacteristic
320C2×C4#39011yescharacteristic
321C2×C4#39211yescharacteristic
322C2×C4#39311yescharacteristic
323C2×C4#39411yescharacteristic
324C2×C4#39411yescharacteristic
325C2×C4#39611yescharacteristic
326C2×C4#39911yescharacteristic
327C2×C4#47112yes—
328C2×C4#47112yes—
329C2×C4#47212yes—
330C2×C4#47312yes—
331C8—44no—
332C8—82no—
333C8—82no—
334C8—82no—
335C8—82no—
336C8—81no—
337C8—81no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—44no—
346C22#143911yescharacteristic
347C22#193112yes—
348C22#206412yes—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—84no—
357C4#190212yes—
358C4#197711yescharacteristic
359C4#197911yescharacteristic
360C4#201811yescharacteristic
361C4#201911yescharacteristic
362C4#205212yes—
363C4#206312yes—
364C2—22no—
365C2—22no—
366C2—41no—
367C2#1105911yescharacteristic
368C2#1112811yescharacteristic
369C2#1114411yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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
#178172
#177422
C2×#15482
C2×#15462
#80042
#79342
#78842