Finite Group Databasebeta

← #117256

Subgroups — #117256

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17817C211yesmaximal, characteristic
2#17746C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4C2×#1549C211yesmaximal, 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#1558—22no—
20#1549—22no—
21#1549—22no—
22#1549—22no—
23#1549—22no—
24#1548—22no—
25#1548—21no—
26#1548—21no—
27C2×#332—22no—
28C2×#332C414yes—
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#423D412yes—
75#423D411yescharacteristic
76#423D411yescharacteristic
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—24no—
94#333—24no—
95#332—28no—
96#332—28no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—84no—
104C23×C4—41no—
105C23×C4#4111yescharacteristic
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—82no—
116C2×M16—82no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—41no—
121C4⋊Q8—22no—
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—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8C4⋊C414yes—
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—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C42C22×C411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—22no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—22no—
173C4⋊C8—41no—
174C4⋊C8—41no—
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—41no—
188#97—41no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—41no—
193C4×C8—41no—
194C4×C8—41no—
195C24#12611yescharacteristic
196C2×Q8—24no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—84no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×Q8—84no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—84no—
215C22×C4#11711yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12611yescharacteristic
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—82no—
238M16—82no—
239M16—82no—
240M16—82no—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C42C2×C22⋊C411yescharacteristic
261C42C2×C4⋊C411yescharacteristic
262C42C4×D411yescharacteristic
263C42C4×D411yescharacteristic
264C42#12011yescharacteristic
265C42#12011yescharacteristic
266C42#12011yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42#12111yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42#12311yescharacteristic
274C42#12411yescharacteristic
275C42#12711yescharacteristic
276C23—22no—
277C23—22no—
278C23—41no—
279C23#38811yescharacteristic
280C23#39611yescharacteristic
281C23#39711yescharacteristic
282Q8—84no—
283Q8—84no—
284Q8—84no—
285Q8—84no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
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—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4#38011yescharacteristic
311C2×C4#38511yescharacteristic
312C2×C4#38511yescharacteristic
313C2×C4#38611yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#39011yescharacteristic
316C2×C4#39211yescharacteristic
317C2×C4#39311yescharacteristic
318C2×C4#39411yescharacteristic
319C2×C4#39411yescharacteristic
320C2×C4#39611yescharacteristic
321C2×C4#39911yescharacteristic
322C2×C4#47112yes—
323C2×C4#47112yes—
324C2×C4#47312yes—
325C2×C4#47312yes—
326C8—44no—
327C8—82no—
328C8—82no—
329C8—82no—
330C8—82no—
331C8—81no—
332C8—81no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—44no—
341C22#143911yescharacteristic
342C22#193012yes—
343C22#206412yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—84no—
352C4#190312yes—
353C4#197711yescharacteristic
354C4#197911yescharacteristic
355C4#201911yescharacteristic
356C4#202011yescharacteristic
357C4#205212yes—
358C4#206412yes—
359C2—22no—
360C2—22no—
361C2—41no—
362C2#1105811yescharacteristic
363C2#1112811yescharacteristic
364C2#1114511yescharacteristic
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
#177462
C2×#15492
#80042
#79342
#78842