Finite Group Databasebeta

← #110792

Subgroups — #110792

Order 512.

627 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#20727C211yesmaximal, characteristic
2#17449C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4#17324C211yesmaximal, characteristic
5#8128C211yesmaximal, characteristic
6#8078C211yesmaximal, characteristic
7#7965C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#1821C412yes—
16#1801—22no—
17#1784C412yes—
18C4×#101—22no—
19#1491C414yes—
20#1487—22no—
21#1487—22no—
22#1487C414yes—
23#1479—22no—
24C2×Q8⋊C8—21no—
25C2×Q8⋊C8—21no—
26#1454—22no—
27#1453—22no—
28#1451—21no—
29C8×M16—21no—
30#1309—22no—
31#1301—22no—
32#1287—22no—
33C23×Q8C2×C411yescharacteristic
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×Q8—21no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C42—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
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#431—21no—
67#431—21no—
68#431C2×C412yes—
69#431D412yes—
70#431D412yes—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423—21no—
76#423C2×C412yes—
77#423D412yes—
78#423D412yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×#101—24no—
83#407—21no—
84#407—21no—
85#407C2×C412yes—
86#407C2×C412yes—
87C2×C22⋊C8—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16C2×C412yes—
91C4×M16D412yes—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94#343—24no—
95Q8⋊C8—28no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98#322—22no—
99#322—24no—
100C82—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—44no—
107C22×Q8C4212yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C22×C411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—41no—
118C2×M16—82no—
119C2×M16C22⋊C414yes—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—41no—
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×Q8—41no—
138C4×Q8—41no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4C4212yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8C22⋊C414yes—
183#101—48no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—41no—
198C4×C8—22no—
199C4×C8—21no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—21no—
203C4×C8—41no—
204C4×C8C22⋊C412yes—
205C4×C8C22⋊C412yes—
206C24#11711yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—81no—
231C22×C4—82no—
232C22×C4C2×C4211yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12611yescharacteristic
249C22×C4#12611yescharacteristic
250M16—42no—
251M16—42no—
252M16—44no—
253M16—44no—
254M16—82no—
255M16—82no—
256M16—82no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—81no—
265C2×C8—81no—
266C2×C8—82no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42C2×C22⋊C411yescharacteristic
280C42C2×C22⋊C411yescharacteristic
281C42#11711yescharacteristic
282C42#11711yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42#12011yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42#12311yescharacteristic
293C42#12411yescharacteristic
294C42#12711yescharacteristic
295C23—22no—
296C23—41no—
297C23—41no—
298C23C4×C22⋊C411yescharacteristic
299C23#38811yescharacteristic
300C23#38811yescharacteristic
301Q8—24no—
302Q8—24no—
303Q8—44no—
304Q8—42no—
305Q8—42no—
306Q8—44no—
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—24no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4#38611yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#39011yescharacteristic
345C2×C4#39011yescharacteristic
346C2×C4#39011yescharacteristic
347C2×C4#47112yes—
348C2×C4#47312yes—
349C8—42no—
350C8—42no—
351C8—44no—
352C8—44no—
353C8—82no—
354C8—82no—
355C8—82no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22#143711yescharacteristic
366C22#193012yes—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—44no—
377C4—82no—
378C4#189112yes—
379C4#190812yes—
380C4#197711yescharacteristic
381C4#197911yescharacteristic
382C2—22no—
383C2—41no—
384C2—41no—
385C2#1093711yescharacteristic
386C2#1094211yescharacteristic
387C2#1095511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#117
LCSG ▷ C22×C4 ▷ 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
#207272
#174492
C2×#14872
#173242
#81282
#80782
#79652