Finite Group Databasebeta

← #161865

Subgroups — #161865

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35187C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7733C212yesmaximal
5#7717C212yesmaximal
6#7713C212yesmaximal
7#7697C212yesmaximal
8#7588C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×#446—21no—
13#2864C2212yes—
14#2863C2212yes—
15#2655—22no—
16#2582C411yescharacteristic
17#2571C411yescharacteristic
18#2568C411yescharacteristic
19#2546C411yescharacteristic
20#2524—22no—
21#2514—21no—
22#2514C412yes—
23#2508C412yes—
24#2446—21no—
25#2327—21no—
26#2320—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2295C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1939—21no—
34#1916—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2212yes—
38#1831C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60#446—44no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×#97—22no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71#401—22no—
72#400—22no—
73#400—22no—
74#400—22no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C411yescharacteristic
87#391C2311yescharacteristic
88#389—21no—
89#389—21no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C411yescharacteristic
96#389C2312yes—
97#389C2311yescharacteristic
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—21no—
108#387—21no—
109#384C2×C411yescharacteristic
110#383—22no—
111#383—22no—
112#382—21no—
113#382—22no—
114#382C2×C412yes—
115#382C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126#376—21no—
127C43C2311yescharacteristic
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C4⋊Q8—24no—
151C4⋊Q8—44no—
152C4×Q8—24no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—24no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2411yescharacteristic
197C4⋊C8—42no—
198C4⋊C8—44no—
199C4⋊C8—44no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—82no—
203#97—44no—
204C4×C8—42no—
205C4×C8—42no—
206C4×C8—81no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42—21no—
303C42—22no—
304C42—21no—
305C42—22no—
306C42—22no—
307C42—41no—
308C42—42no—
309C42—41no—
310C42—42no—
311C42—42no—
312C42—41no—
313C42—42no—
314C42—44no—
315C42—42no—
316C42#13114yes—
317C42AGL(1,ℤ/8)12yes—
318C42#13712yes—
319C23C22×C22⋊C411yescharacteristic
320Q8—44no—
321Q8—82no—
322Q8—84no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4#40812yes—
367C2×C4#40812yes—
368C2×C4#41712yes—
369C2×C4#41712yes—
370C2×C4#42112yes—
371C2×C4C2×#13112yes—
372C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
373C2×C4C2×#13711yescharacteristic
374C2×C4#57711yescharacteristic
375C2×C4#57811yescharacteristic
376C8—42no—
377C8—81no—
378C8—82no—
379C8—81no—
380C8—81no—
381C8—82no—
382C22C2×#40811yescharacteristic
383C22#288411yescharacteristic
384C22#288911yescharacteristic
385C22C2×#41711yescharacteristic
386C22#289811yescharacteristic
387C22C2×#42111yescharacteristic
388C22#290111yescharacteristic
389C4—22no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—41no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—81no—
402C4—81no—
403C4—82no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4#149514yes—
408C2#1739111yescharacteristic
409C2#1746911yescharacteristic
410C2C2×#149511yescharacteristic
411C2#1760111yescharacteristic
412C2#1762811yescharacteristic
413C2#1768311yescharacteristic
414C2#1776311yescharacteristic
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
#351872
#227362
#220182
#77332
#77172
#77132
#76972
#75882
#75772
#75702
#75622