Finite Group Databasebeta

← #164979

Subgroups — #164979

Order 512.

885 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36271C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic, complemented
3#22734C211yesmaximal, characteristic, complemented
4#7732C212yesmaximal, complemented
5#7686C212yesmaximal, complemented
6#7659C211yesmaximal, characteristic
7#7659C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic, complemented
9#7614C212yesmaximal, complemented
10#7574C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12C2×#556—21no—
13C2×#446—21no—
14C2×C8×Q8—21no—
15#2860C2212yes—
16#2858C2211yescharacteristic
17#2601—22no—
18#2592—21no—
19#2560—22no—
20#2548C411yescharacteristic
21#2546C411yescharacteristic
22#2545—21no—
23#2512C412yes—
24#2452C412yes—
25#2367—21no—
26#2366C411yescharacteristic
27#2365C411yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29#2293C2212yes—
30C42×Q8C2211yescharacteristic
31#1940—22no—
32#1939—22no—
33#1939—21no—
34#1939—21no—
35#1831C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1831C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53#556—44no—
54#556—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#126—22no—
60C2×#126—22no—
61C2×#124—21no—
62C2×#124—21no—
63C2×#124—21no—
64C2×C22⋊Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×#117—22no—
73#446—44no—
74C8×Q8—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×#97—22no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#398—22no—
88#397—22no—
89#396—22no—
90#396—22no—
91#394—22no—
92#393—22no—
93#391—21no—
94#391—21no—
95#391—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390C2×C412yes—
101#390C2×C412yes—
102#390C2×C412yes—
103#390C2×C412yes—
104#390C2312yes—
105#389—21no—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2311yescharacteristic
110#389C2311yescharacteristic
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#387—22no—
118#386—22no—
119#385—22no—
120#384C2×C411yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C22⋊C4—22no—
132C43C2311yescharacteristic
133C43C2312yes—
134#336—22no—
135#336—21no—
136#336—22no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8C22⋊C411yescharacteristic
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C23×C4—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—41no—
159C4⋊Q8—42no—
160C4⋊Q8—44no—
161#127—42no—
162#126—48no—
163#126—44no—
164#126—44no—
165#124—44no—
166#124—42no—
167#124—42no—
168#124—42no—
169#124—42no—
170C22⋊Q8—48no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—44no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8D814yes—
178C4×Q8SD1614yes—
179C4×D4—24no—
180C4×D4—42no—
181C4×D4—44no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C22×C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2411yescharacteristic
231C4⋊C8—44no—
232C4⋊C8—44no—
233C4⋊C8—42no—
234C4⋊C8—44no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—44no—
238C4⋊C8—42no—
239#97—44no—
240C4×C8—44no—
241C4×C8—42no—
242C4×C8—42no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—81no—
263C2×Q8#10214yes—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—81no—
332C4⋊C4#10214yes—
333C4⋊C4#10218yes—
334C4⋊C4#13112yes—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C42—22no—
349C42—21no—
350C42—22no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—41no—
356C42—42no—
357C42—41no—
358C42—42no—
359C42—42no—
360C42—41no—
361C42—44no—
362C42—42no—
363C42—42no—
364C42C2×D812yes—
365C42C2×D814yes—
366C42C2×SD1614yes—
367C42C2×SD1612yes—
368C42AGL(1,ℤ/8)12yes—
369C42#13712yes—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23C22×C22⋊C411yescharacteristic
378Q8—28no—
379Q8—84no—
380Q8—84no—
381D4—82no—
382D4—82no—
383D4—84no—
384D4—162no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—88no—
424C2×C4#41312yes—
425C2×C4C2×#10214yes—
426C2×C4C2×#10214yes—
427C2×C4C2×#10212yes—
428C2×C4C2×#10212yes—
429C2×C4#41712yes—
430C2×C4#41712yes—
431C2×C4C2×#13111yescharacteristic
432C2×C4#56811yescharacteristic
433C2×C4C22×D811yescharacteristic
434C2×C4C22×SD1611yescharacteristic
435C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
436C2×C4C2×#13711yescharacteristic
437C8—42no—
438C8—42no—
439C8—81no—
440C8—82no—
441C8—82no—
442C8—81no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22#288011yescharacteristic
451C22#288411yescharacteristic
452C22C2×#41311yescharacteristic
453C22C22×#10211yescharacteristic
454C22C2×#41711yescharacteristic
455C22#290111yescharacteristic
456C22#290111yescharacteristic
457C4—21no—
458C4—21no—
459C4—24no—
460C4—24no—
461C4—22no—
462C4—22no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—41no—
469C4—42no—
470C4—42no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4#154714yes—
476C2—162no—
477C2#1742811yescharacteristic
478C2#1750711yescharacteristic
479C2#1751511yescharacteristic
480C2#1761811yescharacteristic
481C2#1762811yescharacteristic
482C2#1769111yescharacteristic
483C2C2×#154711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22736C2semidirect
3. #22734C2semidirect
4. #7732C2semidirect
5. #7686C2semidirect
8. #7652C2semidirect
9. #7614C2semidirect
10. #7574C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#362712
#227362
#227342
#77322
#76862
#76592
#76522
#76142
#75742
#75702