Finite Group Databasebeta

← #224103

Subgroups — #224103

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34951C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic
3#21339C211yesmaximal, characteristic
4#9875C212yesmaximal
5#9870C211yesmaximal, characteristic
6#9866C212yesmaximal
7#9865C211yesmaximal, characteristic
8#8570C212yesmaximal
9#8564C211yesmaximal, characteristic
10#8182C212yesmaximal
11C4×Q8⋊C8C211yesmaximal, characteristic
12#2585—22no—
13#2582C411yescharacteristic
14#2568C411yescharacteristic
15#2552C411yescharacteristic
16#2546C411yescharacteristic
17#2511—21no—
18#2509C412yes—
19#2453C412yes—
20#2446—21no—
21#2402—22no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#125C2212yes—
25#2295—21no—
26#2295C2212yes—
27C42×Q8C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1916—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35#1831C2212yes—
36C8×C4⋊C4—21no—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77#400—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#391C2×C412yes—
91#391C2×C412yes—
92#389—22no—
93#389—22no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#384—21no—
109#384C2×C411yescharacteristic
110#382C2×C412yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2312yes—
128#376—22no—
129C43—21no—
130C43C2312yes—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140Q8⋊C8—24no—
141Q8⋊C8—22no—
142Q8⋊C8Q814yescomplemented
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22×C411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—44no—
155#125—24no—
156#125—42no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—41no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8C2×Q812yes—
171C4×Q8#4112yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C411yescharacteristic
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×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—22no—
221C4⋊C8—42no—
222C4⋊C8—44no—
223C4⋊C8—42no—
224C4⋊C8—81no—
225C4⋊C8C2×Q812yes—
226C4×C8—21no—
227C4×C8—22no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—22no—
231C4×C8—42no—
232C4×C8—42no—
233C4×C8—42no—
234C4×C8—82no—
235C4×C8C2×Q812yes—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—22no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8C4×Q812yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
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—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—81no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4C4×Q812yes—
330C4⋊C4#13112yes—
331C42—22no—
332C42—22no—
333C42—21no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—21no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—22no—
345C42—22no—
346C42—41no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—41no—
353C42—41no—
354C42—42no—
355C42—42no—
356C42—81no—
357C42#13112yes—
358C42#13112yes—
359C42#13512yes—
360C42#13512yes—
361C42C22×Q811yescharacteristic
362C42C2×#4111yescharacteristic
363C422−1+411yescharacteristic
364C422−1+411yescharacteristic
365C23C22×C22⋊C411yescharacteristic
366Q8—22no—
367Q8—22no—
368Q8—41no—
369Q8—41no—
370Q8—44no—
371Q8—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4#40812yes—
412C2×C4#40812yes—
413C2×C4#41612yes—
414C2×C4#41612yes—
415C2×C4#42112yes—
416C2×C4C2×C4×Q811yescharacteristic
417C2×C4#51911yescharacteristic
418C2×C4#54811yescharacteristic
419C2×C4#54811yescharacteristic
420C2×C4D4×Q811yescharacteristic
421C2×C4D4×Q811yescharacteristic
422C2×C4C2×#13111yescharacteristic
423C2×C4C2×#13111yescharacteristic
424C2×C4C2×#13111yescharacteristic
425C2×C4#56811yescharacteristic
426C2×C4C2×#13511yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57811yescharacteristic
430C8—42no—
431C8—42no—
432C8—42no—
433C8—82no—
434C8—81no—
435C8—82no—
436C22Q8×C22⋊C411yescharacteristic
437C22C2×#40811yescharacteristic
438C22#288011yescharacteristic
439C22C2×#41611yescharacteristic
440C22#289811yescharacteristic
441C22C2×#42111yescharacteristic
442C22#290111yescharacteristic
443C4—21no—
444C4—21no—
445C4—22no—
446C4—22no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—41no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—81no—
457C4—82no—
458C4#148214yes—
459C4#296011yescharacteristic
460C4#296211yescharacteristic
461C4#298511yescharacteristic
462C4#298611yescharacteristic
463C4#336411yescharacteristic
464C4#336511yescharacteristic
465C4#338111yescharacteristic
466C4#338311yescharacteristic
467C2C2×#148211yescharacteristic
468C2#1746911yescharacteristic
469C2#1751011yescharacteristic
470C2#1752311yescharacteristic
471C2#2250911yescharacteristic
472C2#2252111yescharacteristic
473C2#2252911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socle, prime radicalC23C22×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
142. Q8⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349512
#228212
#213392
#98752
#98702
#98662
#98652
#85702
#85642
#81822
C4×Q8⋊C82