Finite Group Databasebeta

← #224165

Subgroups — #224165

Order 512.

761 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36565C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic
3Q8×#97C211yesmaximal, characteristic
4#9895C212yesmaximal
5#9891C211yesmaximal, characteristic
6#9891C211yesmaximal, characteristic
7#9890C212yesmaximal
8#8582C212yesmaximal
9#8580C211yesmaximal, characteristic
10#8227C212yesmaximal
11C4×#330C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2862C2212yes—
14#2859C2211yescharacteristic
15#2586—22no—
16#2586—22no—
17#2551C411yescharacteristic
18#2551C411yescharacteristic
19#2550C412yes—
20#2550C412yes—
21#2451—21no—
22#2451—21no—
23#2369—21no—
24#2369—21no—
25#2355C411yescharacteristic
26#2355C411yescharacteristic
27C4×#125C2211yescharacteristic
28C4×#125C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1770—22no—
34#1770—21no—
35#1767C2211yescharacteristic
36#1767C2212yes—
37#1767C2211yescharacteristic
38#1767C2212yes—
39#1745C2212yes—
40C4×#97C2211yescharacteristic
41C4×#97C2211yescharacteristic
42C4×#97C2212yes—
43C2×#330C2211yescharacteristic
44C2×#330C2212yes—
45C2×#330C2211yescharacteristic
46#1295C2211yescharacteristic
47#1295C2211yescharacteristic
48#1295C2212yes—
49#1295C2212yes—
50#1295C2212yes—
51#1295C2212yes—
52#1295C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2×C412yes—
57C2×#125C2×C412yes—
58C2×#125C2311yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C8×Q8—28no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×#97—21no—
68C2×#97—22no—
69C2×#97—21no—
70C2×#97C2311yescharacteristic
71C2×#97C2311yescharacteristic
72C2×#97C2312yes—
73C2×#97C2312yes—
74C2×#97C2311yescharacteristic
75C2×#97C2311yescharacteristic
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78C2×C4×C8—21no—
79#398—22no—
80#398—22no—
81#395—22no—
82#395—22no—
83#395—22no—
84#395—22no—
85#391—21no—
86#391—21no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391—22no—
91#391—22no—
92#391—22no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—22no—
100#389—22no—
101#389C2×C411yescharacteristic
102#389C2×C411yescharacteristic
103#389C2×C412yes—
104#389C2×C412yes—
105#387—22no—
106#387—22no—
107#387—21no—
108#387—21no—
109#387—22no—
110#387—22no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#382C2×C412yes—
114#382C2×C412yes—
115#382C2312yes—
116#382C2311yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C43C2311yescharacteristic
134C43C2312yes—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#330—22no—
142#330—24no—
143#330Q814yescomplemented
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C22⋊C411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—44no—
154C4⋊Q8—44no—
155C4⋊Q8—82no—
156C4⋊Q8—82no—
157#125—21no—
158#125—21no—
159#125—82no—
160#125C2×Q812yes—
161#125#4112yes—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—24no—
210C4⋊C8—42no—
211C4⋊C8—44no—
212#97—21no—
213#97—21no—
214#97—22no—
215#97—22no—
216#97—21no—
217#97—21no—
218#97—21no—
219#97—21no—
220#97—22no—
221#97—22no—
222#97—44no—
223#97—42no—
224#97—42no—
225#97C2×Q812yes—
226#97C2×Q812yes—
227C4×C8—28no—
228C4×C8—24no—
229C4×C8—42no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×Q8#10114yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—81no—
315C4⋊C4—81no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4#10414yes—
321C4⋊C4C4×Q812yes—
322C4⋊C4C4×Q812yes—
323C42—24no—
324C42—22no—
325C42—24no—
326C42—22no—
327C42—22no—
328C42—42no—
329C42—41no—
330C42—41no—
331C42—42no—
332C42—41no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—82no—
338C42—82no—
339C42C22×Q811yescharacteristic
340C42C2×#4111yescharacteristic
341C422−1+411yescharacteristic
342C422−1+411yescharacteristic
343C23C22×C22⋊C411yescharacteristic
344Q8—28no—
345Q8—164no—
346Q8—164no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4C2×#10114yes—
391C2×C4C2×#10112yes—
392C2×C4C2×#10414yes—
393C2×C4C2×#10412yes—
394C2×C4#42112yes—
395C2×C4C2×C4×Q811yescharacteristic
396C2×C4#51911yescharacteristic
397C2×C4#54811yescharacteristic
398C2×C4#54811yescharacteristic
399C2×C4D4×Q811yescharacteristic
400C2×C4D4×Q811yescharacteristic
401C8—24no—
402C8—44no—
403C8—42no—
404C8—42no—
405C8—42no—
406C8—82no—
407C22Q8×C22⋊C411yescharacteristic
408C22C22×#10111yescharacteristic
409C22#288811yescharacteristic
410C22C22×#10411yescharacteristic
411C22C2×#42111yescharacteristic
412C22#290111yescharacteristic
413C22#290111yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—24no—
419C4—24no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—42no—
424C4—41no—
425C4—42no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4—81no—
431C4—82no—
432C4—81no—
433C4#153114yes—
434C2C2×#153111yescharacteristic
435C2#1765211yescharacteristic
436C2#1768111yescharacteristic
437C2#1769211yescharacteristic
438C2Q8×#10111yescharacteristic
439C2Q8×#10411yescharacteristic
440C2#2252911yescharacteristic
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 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
143. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#365652
#213452
Q8×#972
#98952
#98912
#98902
#85822
#85802
#82272
C4×#3302