Finite Group Databasebeta

← #224155

Subgroups — #224155

Order 512.

835 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36555C211yesmaximal, characteristic
2Q8×#97C211yesmaximal, characteristic, complemented
3#20520C211yesmaximal, characteristic, complemented
4#9892C212yesmaximal, complemented
5#9892C212yesmaximal, complemented
6#9891C211yesmaximal, characteristic, complemented
7#9891C211yesmaximal, characteristic, complemented
8#8581C212yesmaximal
9#8579C211yesmaximal, characteristic
10C4×#329C212yesmaximal
11C4×#329C211yesmaximal, characteristic
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2864C2212yes—
15#2859C2211yescharacteristic
16#2587—22no—
17#2582C411yescharacteristic
18#2538C411yescharacteristic
19#2522—22no—
20#2497—21no—
21#2454C412yes—
22#2453C412yes—
23#2423—21no—
24#2360—21no—
25Q8×C4⋊C4C411yescharacteristic
26D4×C4⋊C4C411yescharacteristic
27#2345—21no—
28C4×#124C2211yescharacteristic
29C4×#124C2212yes—
30C42×Q8C2211yescharacteristic
31#1771—22no—
32#1770—21no—
33#1770—22no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2212yes—
37#1767C2211yescharacteristic
38#1767C2212yes—
39C4×#97C2211yescharacteristic
40C4×#97C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2212yes—
43C2×#329C2212yes—
44C2×#329C2211yescharacteristic
45C2×#329C2211yescharacteristic
46#1295C2212yes—
47#1295C2212yes—
48#1295C2211yescharacteristic
49#1295C2211yescharacteristic
50#1295C2212yes—
51#1295C2212yes—
52#1295C2212yes—
53C2×#125C2×C412yes—
54C2×#124C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C412yes—
63C22×C4⋊C4—21no—
64#446—28no—
65C8×Q8—28no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97C2312yes—
75C2×#97C2312yes—
76C2×#97C2311yescharacteristic
77C2×#97C2311yescharacteristic
78C2×#97C2311yescharacteristic
79C2×#97C2311yescharacteristic
80C2×C4×C8—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—22no—
84#400—22no—
85#399—22no—
86#398—22no—
87#398—22no—
88#396—22no—
89#396—22no—
90#391—21no—
91#391C2×C412yes—
92#391C2×C411yescharacteristic
93#390C2×C411yescharacteristic
94#390C2×C412yes—
95#389—21no—
96#389—22no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—22no—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#388—21no—
107#388—22no—
108#388—22no—
109#388—21no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#385—22no—
116#385—22no—
117#384—21no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382C2312yes—
121#382C2311yescharacteristic
122#380—21no—
123#380—21no—
124#380—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D412yes—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C43C2311yescharacteristic
137C43C2312yes—
138#336—22no—
139#336—22no—
140#336—21no—
141#336—21no—
142#329—24no—
143#329—22no—
144#329Q814yescomplemented
145C22×Q8C22⋊C411yescharacteristic
146C22×Q8C22×C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156#125—82no—
157#124—21no—
158#124—21no—
159#124C2×Q812yes—
160#124#4112yes—
161C22⋊Q8—44no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—82no—
167C4×D4—42no—
168C4×D4—82no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C412yes—
204C2×C42C22×C412yes—
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2411yescharacteristic
215C4⋊C8—28no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—28no—
219#97—28no—
220#97—22no—
221#97—22no—
222#97—21no—
223#97—21no—
224#97—21no—
225#97—21no—
226#97—22no—
227#97—22no—
228#97—21no—
229#97—21no—
230#97—42no—
231#97—44no—
232#97—42no—
233#97C2×Q812yes—
234#97C2×Q812yes—
235C4×C8—28no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—44no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24—21no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8#10414yes—
261C2×Q8C4×Q812yes—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4C4×Q812yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—42no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—82no—
321C4⋊C4—81no—
322C4⋊C4—82no—
323C4⋊C4—81no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4#10414yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C22⋊C4—82no—
338C42—22no—
339C42—24no—
340C42—22no—
341C42—22no—
342C42—24no—
343C42—42no—
344C42—42no—
345C42—41no—
346C42—41no—
347C42—42no—
348C42—42no—
349C42—41no—
350C42—42no—
351C42—82no—
352C42C22×Q811yescharacteristic
353C42C2×#4111yescharacteristic
354C422−1+411yescharacteristic
355C422−1+411yescharacteristic
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C22×C22⋊C411yescharacteristic
364Q8—28no—
365Q8—42no—
366Q8—42no—
367D4—42no—
368D4—42no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—84no—
410C2×C4—164no—
411C2×C4#41312yes—
412C2×C4C2×#10414yes—
413C2×C4C2×#10414yes—
414C2×C4C2×#10412yes—
415C2×C4C2×#10412yes—
416C2×C4C2×C4×Q811yescharacteristic
417C2×C4#51911yescharacteristic
418C2×C4#54811yescharacteristic
419C2×C4#54811yescharacteristic
420C2×C4D4×Q811yescharacteristic
421C2×C4D4×Q811yescharacteristic
422C8—24no—
423C8—24no—
424C8—44no—
425C8—42no—
426C8—42no—
427C8—44no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22Q8×C22⋊C411yescharacteristic
436C22C2×#41311yescharacteristic
437C22#288811yescharacteristic
438C22C22×#10411yescharacteristic
439C22C22×#10411yescharacteristic
440C22#290111yescharacteristic
441C22#290111yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—24no—
447C4—21no—
448C4—21no—
449C4—24no—
450C4—41no—
451C4—42no—
452C4—42no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—81no—
457C4—81no—
458C4—82no—
459C4—82no—
460C4#152414yes—
461C2—82no—
462C2C2×#152411yescharacteristic
463C2#1765211yescharacteristic
464C2#1765511yescharacteristic
465C2#1768711yescharacteristic
466C2#2251511yescharacteristic
467C2Q8×#10411yescharacteristic
468C2Q8×#10411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
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
2. Q8×#97C2semidirect
3. #20520C2semidirect
4. #9892C2semidirect
5. #9892C2semidirect
6. #9891C2semidirect
7. #9891C2semidirect
144. #329Q8semidirect

Maximal subgroups

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

subgroupindex
#365552
Q8×#972
#205202
#98922
#98912
#85812
#85792
C4×#3292