Finite Group Databasebeta

← #6407415

Subgroups — #6407415

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#557C211yesmaximal, characteristic
2#23257C213yesmaximal
3#23254C213yesmaximal
4#23151C213yesmaximal
5#22722C213yesmaximal
6Q8×C4⋊C8C211yesmaximal, characteristic
7Q8×C4⋊C8C211yesmaximal, characteristic
8Q8×C4⋊C8C211yesmaximal, characteristic
9#22303C213yesmaximal
10#22301C213yesmaximal
11#21713C213yesmaximal
12#21340C213yesmaximal
13#20572C211yesmaximal, characteristic
14#20569C211yesmaximal, characteristic
15Q8×#97C211yesmaximal, characteristic
16C2×#557C411yescharacteristic
17C2×#446C2216yes—
18C2×C8×Q8C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22#2370C413yes—
23#2370C413yes—
24#2365C413yes—
25#2355C413yes—
26#2351C411yescharacteristic
27#2351C411yescharacteristic
28Q8×C4⋊C4C411yescharacteristic
29C4×C4⋊Q8C2213yes—
30C4×#125C2213yes—
31C4×#125C2213yes—
32C42×Q8C2211yescharacteristic
33C42×Q8C2211yescharacteristic
34C42×Q8C2211yescharacteristic
35C42×Q8C2213yes—
36#1985C2213yes—
37#1985C2213yes—
38#1940C2216yes—
39#1940C2216yes—
40#1940C2211yescharacteristic
41#1940C2213yes—
42#1939C2213yes—
43#1939C2213yes—
44#1939C2216yes—
45#1939C2213yes—
46#1939C2213yes—
47#1939C2213yes—
48#1939C2213yes—
49#1917C2213yes—
50#1917C2213yes—
51#1917C2216yes—
52#1917C2213yes—
53#1917C2213yes—
54#1916C2216yes—
55#1842C2213yes—
56#1842C2213yes—
57#1842C2213yes—
58#1771C2216yes—
59#1770C2213yes—
60#1770C2213yes—
61#1770C2213yes—
62#1770C2213yes—
63C8×C4⋊C4C2213yes—
64C8×C4⋊C4C2213yes—
65#1767C2213yes—
66#1767C2213yes—
67C4×C4⋊C8C2213yes—
68C4×C4⋊C8C2213yes—
69C4×C4⋊C8C2213yes—
70C4×C4⋊C8C2213yes—
71#1744C2213yes—
72#1744C2213yes—
73C4×#97C2213yes—
74#557—28no—
75C2×C4⋊Q8C2×C413yes—
76C2×#125C2×C413yes—
77C2×#125C2×C413yes—
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C413yes—
84C2×C4×Q8C2×C413yes—
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2313yes—
87C2×C4×Q8C2311yescharacteristic
88#446—412no—
89C8×Q8—24no—
90C8×Q8—24no—
91C8×Q8—24no—
92C8×Q8Q818yescomplemented
93C2×C4⋊C8C2313yes—
94C2×C4⋊C8C2313yes—
95C2×C4⋊C8C2313yes—
96C2×C4⋊C8C2313yes—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2313yes—
101C2×C4⋊C8C2313yes—
102C2×C4⋊C8C2316yes—
103C2×C4⋊C8C2316yes—
104C2×C4⋊C8C2316yes—
105C2×C4⋊C8C2313yes—
106C2×#97C2313yes—
107C2×#97C2313yes—
108C2×#97C2311yescharacteristic
109C2×#97C2313yes—
110C2×#97C2316yes—
111C2×C4×C8C2313yes—
112C2×C4×C8C2313yes—
113C2×C4×C8C2313yes—
114C2×C4×C8C2316yes—
115C2×C4×C8C2313yes—
116#391C2×C416yes—
117#389C2×C413yes—
118#389C2×C413yes—
119#389C2×C413yes—
120#389C2×C413yes—
121#389C2×C413yes—
122#389C2×C413yes—
123#389C2×C416yes—
124#387C2×C416yes—
125#387C2×C416yes—
126#387C2×C416yes—
127#387C2×C416yes—
128#384C2×C413yes—
129#382C2×C413yes—
130#382C2×C413yes—
131C4×C4⋊C4C2×C413yes—
132C4×C4⋊C4C2×C413yes—
133C4×C4⋊C4C2×C413yes—
134C4×C4⋊C4C2×C413yes—
135C4×C4⋊C4C2×C413yes—
136C4×C4⋊C4C2×C413yes—
137C4×C4⋊C4C2×C413yes—
138C4×C4⋊C4C2×C413yes—
139C4×C4⋊C4C2×C413yes—
140C4×C4⋊C4C2×C413yes—
141C4×C4⋊C4C2×C413yes—
142C4×C4⋊C4C2×C413yes—
143C4×C4⋊C4C2316yes—
144C4×C4⋊C4C2313yes—
145C4×C4⋊C4C2313yes—
146C4×C4⋊C4C2313yes—
147C4×C4⋊C4C2313yes—
148C4×C4⋊C4C2313yes—
149C4×C4⋊C4C2311yescharacteristic
150C43C2313yes—
151C43C2313yes—
152C43C2313yes—
153#336C2313yes—
154#336C2313yes—
155#336C2313yes—
156#336C2316yes—
157#336C2313yes—
158#336C2313yes—
159#336C2311yescharacteristic
160#336C2311yescharacteristic
161#336C2313yes—
162#336C2313yes—
163#336C2313yes—
164#336C2313yes—
165#336C2313yes—
166#336C2313yes—
167#336C2313yes—
168C22×Q8C22×C413yes—
169C22×Q8C22×C411yescharacteristic
170C22×C8C2411yescharacteristic
171C22×C8C2411yescharacteristic
172C22×C8C2413yes—
173C22×C8C2413yes—
174C22×C8C2413yes—
175C22×C8C2411yescharacteristic
176C22×C8C2413yes—
177C22×C8C2411yescharacteristic
178C4⋊Q8—212no—
179#125—212no—
180#125—212no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—212no—
187C4×Q8—26no—
188C4×Q8—26no—
189C4×Q8—46no—
190C4×Q8M1618yes—
191C4×Q8C2×Q814yes—
192C4×Q8#4114yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C413yes—
196C2×C4⋊C4C22×C413yes—
197C2×C4⋊C4C22×C413yes—
198C2×C4⋊C4C22×C413yes—
199C2×C4⋊C4C22×C413yes—
200C2×C4⋊C4C22×C413yes—
201C2×C4⋊C4C22×C413yes—
202C2×C4⋊C4C22×C413yes—
203C2×C4⋊C4C22×C413yes—
204C2×C4⋊C4C22×C413yes—
205C2×C4⋊C4C22×C413yes—
206C2×C4⋊C4C22×C416yes—
207C2×C4⋊C4C22×C413yes—
208C2×C4⋊C4C22×C416yes—
209C2×C4⋊C4C22×C413yes—
210C2×C4⋊C4C22×C413yes—
211C2×C42C22×C413yes—
212C2×C42C22×C413yes—
213C2×C42C22×C413yes—
214C2×C42C22×C413yes—
215C2×C42C22×C413yes—
216C2×C42C22×C413yes—
217C2×C42C22×C413yes—
218C2×C42C22×C413yes—
219C2×C42C22×C413yes—
220C2×C42C22×C413yes—
221C2×C42C22×C413yes—
222C2×C42C22×C413yes—
223C2×C42C2413yes—
224C2×C42C2413yes—
225C2×C42C2411yescharacteristic
226C2×C42C2411yescharacteristic
227C2×C42C2413yes—
228C2×C42C2413yes—
229C2×C42C2411yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—26no—
233C4⋊C8—26no—
234C4⋊C8—26no—
235C4⋊C8—26no—
236C4⋊C8—22no—
237C4⋊C8—46no—
238C4⋊C8—43no—
239C4⋊C8—43no—
240C4⋊C8—46no—
241C4⋊C8—46no—
242C4⋊C8C2×Q8112yes—
243#97—26no—
244#97—26no—
245#97—22no—
246#97—43no—
247#97—46no—
248C4×C8—26no—
249C4×C8—26no—
250C4×C8—26no—
251C4×C8—46no—
252C4×C8C2×Q8112yes—
253#95C22×C411yescharacteristic
254#95C22×C411yescharacteristic
255#95C22×C413yes—
256#95C22×C413yes—
257#95C22×C416yes—
258#95C22×C413yes—
259#95C22×C413yes—
260#95C22×C416yes—
261#95C22×C416yes—
262#95C22×C416yes—
263#95C22×C416yes—
264C2×Q8—26no—
265C2×Q8—26no—
266C2×Q8—43no—
267C2×Q8C4×Q814yes—
268C2×Q8C2×M1614yes—
269C2×Q8#13114yes—
270C22×C4C23×C413yes—
271C22×C4C23×C413yes—
272C22×C4C23×C413yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C23×C413yes—
276C22×C4C23×C413yes—
277C22×C4C23×C413yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C23×C413yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C23×C413yes—
284C22×C4C2511yescharacteristic
285C2×C8—21no—
286C2×C8—23no—
287C2×C8—21no—
288C2×C8—23no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—23no—
294C2×C8—23no—
295C2×C8—23no—
296C2×C8—23no—
297C2×C8—23no—
298C2×C8—23no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—23no—
304C2×C8—23no—
305C2×C8—23no—
306C2×C8C22×Q812yes—
307C2×C8C22×Q816yes—
308C2×C82−1+412yes—
309C4⋊C4—26no—
310C4⋊C4—26no—
311C4⋊C4—26no—
312C4⋊C4—22no—
313C4⋊C4—26no—
314C4⋊C4—26no—
315C4⋊C4—26no—
316C4⋊C4—26no—
317C4⋊C4—212no—
318C4⋊C4—26no—
319C4⋊C4—26no—
320C4⋊C4—26no—
321C4⋊C4—26no—
322C4⋊C4—22no—
323C4⋊C4—46no—
324C4⋊C4—43no—
325C4⋊C4C4×Q8112yes—
326C4⋊C4C2×M16112yes—
327C42—26no—
328C42—26no—
329C42—26no—
330C42—26no—
331C42—26no—
332C42—26no—
333C42—23no—
334C42—23no—
335C42—23no—
336C42—21no—
337C42—21no—
338C42—23no—
339C42—23no—
340C42—23no—
341C42—21no—
342C42—21no—
343C42—43no—
344C42—43no—
345C42—43no—
346C42—43no—
347C42—43no—
348C42C2×M16112yes—
349C42C22×Q816yes—
350C42C2×#4116yes—
351C42H(5,𝔽2)12yes—
352C422−1+412yes—
353C23C24×C411yescharacteristic
354Q8—412no—
355Q8#446116yescomplemented
356C2×C4—23no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—23no—
363C2×C4—21no—
364C2×C4—23no—
365C2×C4—23no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—23no—
369C2×C4—23no—
370C2×C4—23no—
371C2×C4—23no—
372C2×C4—21no—
373C2×C4—23no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—23no—
377C2×C4—23no—
378C2×C4—21no—
379C2×C4—23no—
380C2×C4—23no—
381C2×C4—23no—
382C2×C4—23no—
383C2×C4—21no—
384C2×C4—23no—
385C2×C4—21no—
386C2×C4—23no—
387C2×C4—23no—
388C2×C4—23no—
389C2×C4C2×C4×Q816yes—
390C2×C4C2×C4×Q816yes—
391C2×C4#52012yes—
392C2×C4#52012yes—
393C2×C4C22×M1612yes—
394C2×C4C22×M1616yes—
395C2×C4C22×M1616yes—
396C2×C4C2×#13116yes—
397C2×C4#56812yes—
398C2×C4C23×Q811yescharacteristic
399C2×C4C22×#4111yescharacteristic
400C2×C4C2×H(5,𝔽2)11yescharacteristic
401C2×C4C2×2−1+411yescharacteristic
402C2×C4C2×2−1+411yescharacteristic
403C2×C4#58511yescharacteristic
404C8—22no—
405C8—26no—
406C8—22no—
407C8—43no—
408C8—43no—
409C8—43no—
410C8—41no—
411C8—41no—
412C22C22×C4×Q811yescharacteristic
413C22C2×#52011yescharacteristic
414C22C4×2−1+411yescharacteristic
415C22C23×M1611yescharacteristic
416C22C22×#13111yescharacteristic
417C22C2×#56811yescharacteristic
418C22#357311yescharacteristic
419C4—26no—
420C4—22no—
421C4—26no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—43no—
426C4—43no—
427C4—43no—
428C4—41no—
429C4—43no—
430C4—41no—
431C4—43no—
432C4C2×#446112yes—
433C4#350211yescharacteristic
434C4#350311yescharacteristic
435C4#351711yescharacteristic
436C4#352311yescharacteristic
437C2#3433711yescharacteristic
438C2C22×#44611yescharacteristic
439C2#6019511yescharacteristic
440C2#6026311yescharacteristic
441C2#6027411yescharacteristic
442C2#6033311yescharacteristic
443C2#6033511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socle, prime radicalC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
92. C8×Q8Q8semidirect
355. Q8#446semidirect

Maximal subgroups

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

subgroupindex
C4×#5572
#232572
#232542
#231512
#227222
Q8×C4⋊C82
#223032
#223012
#217132
#213402
#205722
#205692
Q8×#972