Finite Group Databasebeta

← #164992

Subgroups — #164992

Order 512.

817 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36466C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic
3#22736C211yesmaximal, characteristic
4#7723C212yesmaximal
5#7680C212yesmaximal
6#7663C211yesmaximal, characteristic
7#7660C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic
9#7625C212yesmaximal
10#7588C212yesmaximal
11#7570C211yesmaximal, characteristic
12C2×#557—21no—
13C2×#446—21no—
14#2859C2212yes—
15#2859C2211yescharacteristic
16#2584—21no—
17#2584—22no—
18#2552C411yescharacteristic
19#2552C412yes—
20#2551C412yes—
21#2551C411yescharacteristic
22#2550—22no—
23#2550—21no—
24#2351—21no—
25Q8×C4⋊C4C411yescharacteristic
26Q8×C4⋊C4C411yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1916—22no—
34#1839C2212yes—
35#1831C2212yes—
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42#1744C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53#557—44no—
54#557—44no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×#125—22no—
60C2×#125—21no—
61C2×#125—21no—
62C2×#125—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70#446—44no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×#97—22no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#398—22no—
82#398—22no—
83#398—22no—
84#395—22no—
85#395—22no—
86#395—22no—
87#391—22no—
88#391—21no—
89#389—22no—
90#389—21no—
91#389—22no—
92#389—21no—
93#389—21no—
94#389—22no—
95#389—21no—
96#389—22no—
97#389—21no—
98#389—22no—
99#389—22no—
100#389—21no—
101#389—22no—
102#389—21no—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2311yescharacteristic
107#389C2311yescharacteristic
108#389C2312yes—
109#384C2×C411yescharacteristic
110#384C2×C412yes—
111#384C2×C412yes—
112#384C2×C411yescharacteristic
113#384C2×C412yes—
114#384C2×C411yescharacteristic
115#382—22no—
116#382—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C43C2311yescharacteristic
130C43C2312yes—
131#336—21no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—41no—
155C4⋊Q8—42no—
156C4⋊Q8—48no—
157C4⋊Q8—44no—
158#125—48no—
159#125—44no—
160#125—42no—
161#125—44no—
162#125—44no—
163#125—42no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—44no—
168C4×Q8—44no—
169C4×Q8—42no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—42no—
173C4×Q8D814yes—
174C4×Q8SD1614yes—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—42no—
225C4⋊C8—44no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—44no—
229C4⋊C8—42no—
230#97—44no—
231C4×C8—44no—
232C4×C8—42no—
233C4×C8—42no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—42no—
250C2×Q8—44no—
251C2×Q8—42no—
252C2×Q8—44no—
253C2×Q8—82no—
254C2×Q8—81no—
255C2×Q8#10214yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C4⋊C4—21no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—81no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4#10114yes—
326C4⋊C4#10218yes—
327C4⋊C4#10214yes—
328C4⋊C4#13112yes—
329C42—21no—
330C42—22no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—22no—
337C42—42no—
338C42—42no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—44no—
343C42—42no—
344C42—44no—
345C42—42no—
346C42C2×D812yes—
347C42C2×D814yes—
348C42C2×SD1614yes—
349C42C2×SD1612yes—
350C42#13712yes—
351C42#13712yes—
352C23C22×C22⋊C411yescharacteristic
353Q8—28no—
354Q8—88no—
355Q8—84no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4C2×#10114yes—
397C2×C4C2×#10112yes—
398C2×C4C2×#10214yes—
399C2×C4C2×#10212yes—
400C2×C4C2×#10214yes—
401C2×C4C2×#10212yes—
402C2×C4#41912yes—
403C2×C4#41912yes—
404C2×C4C2×#13111yescharacteristic
405C2×C4#56811yescharacteristic
406C2×C4C22×D811yescharacteristic
407C2×C4C22×SD1611yescharacteristic
408C2×C4C2×#13711yescharacteristic
409C2×C4C2×#13711yescharacteristic
410C8—42no—
411C8—42no—
412C8—81no—
413C8—82no—
414C8—81no—
415C8—82no—
416C22#288011yescharacteristic
417C22#288411yescharacteristic
418C22C22×#10111yescharacteristic
419C22C22×#10211yescharacteristic
420C22C2×#41911yescharacteristic
421C22#290111yescharacteristic
422C22#290111yescharacteristic
423C4—21no—
424C4—21no—
425C4—24no—
426C4—24no—
427C4—24no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—82no—
439C4—81no—
440C4—82no—
441C4—81no—
442C4#154614yes—
443C2#1743011yescharacteristic
444C2#1750711yescharacteristic
445C2#1751411yescharacteristic
446C2#1761811yescharacteristic
447C2#1763211yescharacteristic
448C2#1768911yescharacteristic
449C2C2×#154611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#364662
#228282
#227362
#77232
#76802
#76632
#76602
#76522
#76252
#75882
#75702