Finite Group Databasebeta

← #116200

Subgroups — #116200

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21973C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3C2×#1550C211yesmaximal, characteristic
4#17584C211yesmaximal, characteristic
5#8082C211yesmaximal, characteristic
6#8065C211yesmaximal, characteristic
7#7962C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2283C2211yescharacteristic
15#1939—22no—
16#1850—22no—
17#1843—21no—
18#1832—21no—
19#1552—21no—
20#1552—21no—
21#1550—22no—
22#1550—22no—
23#1549—21no—
24#1546—21no—
25#1544—22no—
26C2×#333—21no—
27C2×#332—21no—
28#1518—22no—
29#1515C414yes—
30#1508—22no—
31#1301—22no—
32#1299—22no—
33#1286—22no—
34C22×M16—21no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431D412yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423D412yes—
71C2×C4⋊C8—21no—
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⋊C8—24no—
79C2×C4⋊C8D412yes—
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82#407—21no—
83#407—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C2×C4×C8—21no—
88#383C2×C412yes—
89#382D412yes—
90#381—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C22⋊C4D412yes—
94#376—21no—
95#336—24no—
96#333—24no—
97#332—24no—
98#332—22no—
99#332—22no—
100#332—22no—
101#332—22no—
102#328—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—22no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—82no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—82no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C4⋊C412yes—
127C4⋊Q8C4⋊C412yes—
128#125—22no—
129#125—22no—
130#125—41no—
131#125—41no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—22no—
150#117—41no—
151#117—41no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×Q811yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—84no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198#97—22no—
199#97—41no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8—41no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95C22⋊C414yes—
208C24#12311yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×Q8—82no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12711yescharacteristic
242M16—82no—
243M16—81no—
244M16—81no—
245M16—82no—
246M16—82no—
247C2×C8—42no—
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—82no—
258C2×C8—82no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42C2×C4⋊C411yescharacteristic
281C42#11711yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42#12011yescharacteristic
285C42#12011yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42#12411yescharacteristic
295C42#12411yescharacteristic
296C23—22no—
297C23—22no—
298C23—41no—
299C23#38611yescharacteristic
300C23#39711yescharacteristic
301C23#39911yescharacteristic
302Q8—82no—
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—21no—
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—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#38011yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38811yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#39011yescharacteristic
342C2×C4#39211yescharacteristic
343C2×C4#39311yescharacteristic
344C2×C4#39411yescharacteristic
345C2×C4#39411yescharacteristic
346C2×C4#39611yescharacteristic
347C2×C4#39611yescharacteristic
348C2×C4#47112yes—
349C2×C4#47312yes—
350C8—82no—
351C8—82no—
352C8—81no—
353C8—82no—
354C8—81no—
355C8—82no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—44no—
364C22#143911yescharacteristic
365C22#190812yes—
366C22#208012yes—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—84no—
377C4—82no—
378C4—82no—
379C4#200811yescharacteristic
380C4#200911yescharacteristic
381C4#204611yescharacteristic
382C4#204711yescharacteristic
383C4#206412yes—
384C2—22no—
385C2—22no—
386C2—41no—
387C2#1106211yescharacteristic
388C2#1108611yescharacteristic
389C2#1109311yescharacteristic
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

Maximal subgroups

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

subgroupindex
#219732
#177152
C2×#15502
#175842
#80822
#80652
#79622