Finite Group Databasebeta

← #224452

Subgroups — #224452

Order 512.

735 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36555C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic, complemented
3#21345C211yesmaximal, characteristic, complemented
4#9893C212yesmaximal, complemented
5#9893C212yesmaximal, complemented
6#9891C211yesmaximal, characteristic, complemented
7#9891C211yesmaximal, characteristic, complemented
8#8581C212yesmaximal
9#8579C211yesmaximal, characteristic
10#8224C212yesmaximal
11C4×#329C211yesmaximal, characteristic
12#2864C2212yes—
13#2859C2211yescharacteristic
14#2587—22no—
15#2582C411yescharacteristic
16#2538C411yescharacteristic
17#2522—22no—
18#2497—21no—
19#2454C412yes—
20#2453C412yes—
21#2423—21no—
22#2360—21no—
23Q8×C4⋊C4C411yescharacteristic
24D4×C4⋊C4C411yescharacteristic
25#2345—21no—
26C4×#124C2211yescharacteristic
27C4×#124C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1767C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2212yes—
38#1767C2212yes—
39#1745C2212yes—
40#1745C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×#329C2211yescharacteristic
44C2×#329C2212yes—
45C2×#329C2211yescharacteristic
46#1295C2211yescharacteristic
47#1295C2211yescharacteristic
48#1295C2212yes—
49#1295C2212yes—
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—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97C2311yescharacteristic
71C2×#97C2311yescharacteristic
72C2×#97C2312yes—
73C2×#97C2312yes—
74C2×#97C2311yescharacteristic
75C2×#97C2311yescharacteristic
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#400—22no—
79#399—22no—
80#398—22no—
81#398—22no—
82#396—22no—
83#396—22no—
84#391—21no—
85#391C2×C412yes—
86#391C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#390C2×C412yes—
89#389—21no—
90#389—22no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—22no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#388—21no—
101#388—22no—
102#388—22no—
103#388—21no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#385—22no—
110#385—22no—
111#384—21no—
112#384C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114#382C2312yes—
115#382C2311yescharacteristic
116#380—21no—
117#380—21no—
118#380—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#329—22no—
141#329—24no—
142#329Q814yescomplemented
143C22×Q8C22⋊C411yescharacteristic
144C22×Q8C22×C411yescharacteristic
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154#125—82no—
155#124—21no—
156#124—21no—
157#124C2×Q812yes—
158#124#4112yes—
159C22⋊Q8—44no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—82no—
165C4×D4—42no—
166C4×D4—82no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C412yes—
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×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—42no—
214C4⋊C8—42no—
215C4⋊C8—44no—
216C4⋊C8—44no—
217#97—21no—
218#97—21no—
219#97—22no—
220#97—22no—
221#97—21no—
222#97—21no—
223#97—21no—
224#97—21no—
225#97—22no—
226#97—22no—
227#97—42no—
228#97—42no—
229#97C2×Q812yes—
230#97C2×Q812yes—
231C4×C8—42no—
232C4×C8—42no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C24—21no—
249C2×Q8—21no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8C4×Q812yes—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4C4×Q812yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—81no—
317C4⋊C4—82no—
318C4⋊C4—81no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—82no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C22⋊C4—82no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—41no—
336C42—41no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—82no—
346C42C22×Q811yescharacteristic
347C42C2×#4111yescharacteristic
348C422−1+411yescharacteristic
349C422−1+411yescharacteristic
350C23—22no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C22×C22⋊C411yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—44no—
361D4—42no—
362D4—42no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—164no—
407C2×C4#41312yes—
408C2×C4#42112yes—
409C2×C4#42112yes—
410C2×C4C2×C4×Q811yescharacteristic
411C2×C4#51911yescharacteristic
412C2×C4#54811yescharacteristic
413C2×C4#54811yescharacteristic
414C2×C4D4×Q811yescharacteristic
415C2×C4D4×Q811yescharacteristic
416C8—42no—
417C8—42no—
418C8—42no—
419C8—42no—
420C8—82no—
421C8—82no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22Q8×C22⋊C411yescharacteristic
430C22C2×#41311yescharacteristic
431C22#288811yescharacteristic
432C22C2×#42111yescharacteristic
433C22C2×#42111yescharacteristic
434C22#290111yescharacteristic
435C22#290111yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—42no—
444C4—42no—
445C4—41no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4#153514yes—
455C2—82no—
456C2C2×#153511yescharacteristic
457C2#1768111yescharacteristic
458C2#1768411yescharacteristic
459C2#1768711yescharacteristic
460C2#2251511yescharacteristic
461C2#2252911yescharacteristic
462C2#2252911yescharacteristic
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. #21345C2semidirect
3. #21345C2semidirect
4. #9893C2semidirect
5. #9893C2semidirect
6. #9891C2semidirect
7. #9891C2semidirect
142. #329Q8semidirect

Maximal subgroups

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

subgroupindex
#365552
#213452
#98932
#98912
#85812
#85792
#82242
C4×#3292