Finite Group Databasebeta

← #323599

Subgroups — #323599

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30307C211yesmaximal, characteristic, complemented
2#29193C211yesmaximal, characteristic, complemented
3#29192C211yesmaximal, characteristic
4#29178C211yesmaximal, characteristic, complemented
5#29143C211yesmaximal, characteristic
6#22219C211yesmaximal, characteristic
7#10679C211yesmaximal, characteristic, complemented
8#10679C211yesmaximal, characteristic, complemented
9#10678C211yesmaximal, characteristic
10#10673C211yesmaximal, characteristic
11#9827C211yesmaximal, characteristic
12#9822C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8264C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17#2859C2211yescharacteristic
18#2859C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×#127C2211yescharacteristic
21#2086—22no—
22#2085—22no—
23#2068—22no—
24#2067—22no—
25#2059—22no—
26#2025—22no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1926—22no—
38#1918—22no—
39#1864—22no—
40#1863—22no—
41#1862—22no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1804—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#127C2311yescharacteristic
65C2×#125—21no—
66C2×#125—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71#486—22no—
72#486Q814yes—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×C4⋊C8D412yes—
80C2×C4⋊C8D412yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#382—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C43C2311yescharacteristic
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#336D412yes—
131#336D412yes—
132#336D412yes—
133#336D412yes—
134#322—22no—
135C82—22no—
136C22×Q8—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C23×C4—41no—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C2×Q812yes—
151#127—21no—
152#127—21no—
153#127—21no—
154#127C2×Q812yes—
155#125—82no—
156#125—82no—
157C4×Q8—82no—
158C4×D4—44no—
159C4×D4—82no—
160C4×D4—82no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C42—41no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2411yescharacteristic
197#107—44no—
198#107—82no—
199#107—82no—
200#107—82no—
201#106—82no—
202#106—82no—
203C4⋊C8—24no—
204C4⋊C8—42no—
205#103—44no—
206#103—82no—
207#102—28no—
208#102—44no—
209#102—82no—
210#102—82no—
211#102—82no—
212#102—82no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8C2×Q812yes—
224C4×C8#4112yes—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24—41no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4#12711yescharacteristic
259C22×C4#12711yescharacteristic
260C22×C4#12711yescharacteristic
261C22×C4#12711yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8C22⋊Q814yes—
282C2×C8#13514yes—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—82no—
320C42C22×Q811yescharacteristic
321C42C2×#4111yescharacteristic
322C422−1+411yescharacteristic
323C422−1+411yescharacteristic
324C23—42no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23C2×#12711yescharacteristic
332Q8—44no—
333Q8—82no—
334D4—44no—
335D4—44no—
336D4—44no—
337D4—82no—
338D4—82no—
339D4—82no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—164no—
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4#53611yescharacteristic
376C2×C4#54711yescharacteristic
377C2×C4#54711yescharacteristic
378C2×C4D4×Q811yescharacteristic
379C2×C4D4×Q811yescharacteristic
380C2×C4C2×#13512yes—
381C2×C4#57512yes—
382C2×C4#57612yes—
383C2×C4#57612yes—
384C2×C4#57712yes—
385C2×C4#57812yes—
386C8—24no—
387C8—24no—
388C8—42no—
389C8—42no—
390C8—42no—
391C8—42no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22#254211yescharacteristic
400C22#314711yescharacteristic
401C22#315311yescharacteristic
402C22#315411yescharacteristic
403C22#315511yescharacteristic
404C22#315511yescharacteristic
405C22#315611yescharacteristic
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—82no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4#308312yes—
418C4#324312yes—
419C4#333712yes—
420C4#334712yes—
421C4#338112yes—
422C4#338912yes—
423C2—162no—
424C2#1964311yescharacteristic
425C2#1966711yescharacteristic
426C2#1971511yescharacteristic
427C2#1974111yescharacteristic
428C2#2852311yescharacteristic
429C2#2853311yescharacteristic
430C2#2854411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30307C2semidirect
2. #29193C2semidirect
4. #29178C2semidirect
7. #10679C2semidirect
8. #10679C2semidirect
13. #8821C2semidirect
14. #8814C2semidirect
15. #8264C2semidirect

Maximal subgroups

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

subgroupindex
#303072
#291932
#291922
#291782
#291432
#222192
#106792
#106782
#106732
#98272
#98222
#88212
#88142
#82642