Finite Group Databasebeta

← #165124

Subgroups — #165124

Order 512.

781 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36466C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic
3#22832C211yesmaximal, characteristic
4#7703C212yesmaximal
5#7700C212yesmaximal
6#7663C211yesmaximal, characteristic
7#7660C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic
9#7638C212yesmaximal
10#7604C212yesmaximal
11#7570C211yesmaximal, characteristic
12C2×#557—21no—
13#2859C2212yes—
14#2859C2211yescharacteristic
15#2584—22no—
16#2584—21no—
17#2552C411yescharacteristic
18#2552C412yes—
19#2551C412yes—
20#2551C411yescharacteristic
21#2550—22no—
22#2550—21no—
23#2351—21no—
24Q8×C4⋊C4C411yescharacteristic
25Q8×C4⋊C4C411yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1917—22no—
32#1917—22no—
33#1839C2212yes—
34#1839C2212yes—
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37#1770—21no—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1745C2212yes—
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—22no—
62C2×#125—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×#97—22no—
75C2×#97—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#398—22no—
79#398—22no—
80#398—22no—
81#395—22no—
82#395—22no—
83#395—22no—
84#391—22no—
85#391—21no—
86#389—22no—
87#389—21no—
88#389—22no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—21no—
93#389—22no—
94#389—22no—
95#389—21no—
96#389—22no—
97#389—21no—
98#389—22no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2311yescharacteristic
104#389C2311yescharacteristic
105#389C2312yes—
106#384C2×C411yescharacteristic
107#384C2×C412yes—
108#384C2×C412yes—
109#384C2×C411yescharacteristic
110#384C2×C412yes—
111#384C2×C411yescharacteristic
112#382—22no—
113#382—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C43C2311yescharacteristic
127C43C2312yes—
128#336—21no—
129#336—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—41no—
154C4⋊Q8—42no—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157#125—44no—
158#125—44no—
159#125—48no—
160#125—44no—
161#125—42no—
162#125—42no—
163C4×Q8—24no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—42no—
173C4×Q8—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C412yes—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—42no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227#97—44no—
228#97—44no—
229C4×C8—42no—
230C4×C8—42no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—81no—
251C2×Q8—82no—
252C2×Q8#10114yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—42no—
288C4⋊C4—21no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
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—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—81no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C4⋊C4—81no—
325C4⋊C4#13112yes—
326C42—21no—
327C42—22no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—24no—
335C42—22no—
336C42—42no—
337C42—42no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—44no—
342C42—44no—
343C42—42no—
344C42—42no—
345C42AGL(1,ℤ/8)12yes—
346C42AGL(1,ℤ/8)12yes—
347C42#13712yes—
348C42#13712yes—
349C23C22×C22⋊C411yescharacteristic
350Q8—28no—
351Q8—84no—
352Q8—88no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4C2×#10114yes—
396C2×C4C2×#10112yes—
397C2×C4#41812yes—
398C2×C4#41812yes—
399C2×C4#41912yes—
400C2×C4#41912yes—
401C2×C4C2×#13111yescharacteristic
402C2×C4#56811yescharacteristic
403C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
404C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4C2×#13711yescharacteristic
407C8—42no—
408C8—42no—
409C8—81no—
410C8—82no—
411C8—81no—
412C8—82no—
413C22#288011yescharacteristic
414C22#288411yescharacteristic
415C22C22×#10111yescharacteristic
416C22C2×#41811yescharacteristic
417C22C2×#41911yescharacteristic
418C22#290111yescharacteristic
419C22#290111yescharacteristic
420C4—21no—
421C4—21no—
422C4—24no—
423C4—22no—
424C4—22no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—41no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4—82no—
438C4—81no—
439C4#155614yes—
440C2#1743011yescharacteristic
441C2#1751111yescharacteristic
442C2#1751411yescharacteristic
443C2#1763111yescharacteristic
444C2#1763211yescharacteristic
445C2#1769211yescharacteristic
446C2C2×#155611yescharacteristic
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
#228322
#77032
#77002
#76632
#76602
#76522
#76382
#76042
#75702