Finite Group Databasebeta

← #220367

Subgroups — #220367

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3299C211yesmaximal, characteristic
2#22515C211yesmaximal, characteristic
3Q8×#100C211yesmaximal, characteristic
4#18042C212yesmaximal, complemented
5#9803C212yesmaximal, complemented
6#9250C212yesmaximal, complemented
7#9248C212yesmaximal, complemented
8#8614C211yesmaximal, characteristic
9#8395C211yesmaximal, characteristic
10#8391C211yesmaximal, characteristic
11#8164C211yesmaximal, characteristic
12C2×#554C2211yescharacteristic
13C2×D4×Q8C2211yescharacteristic
14#3299—22no—
15#3299—22no—
16#3299—22no—
17#3299—22no—
18C2×#487C2211yescharacteristic
19C2×#484C412yes—
20C2×#477C2211yescharacteristic
21C2×#467C412yes—
22C2×#462C2211yescharacteristic
23C2×#451C412yes—
24Q8×M16C2212yes—
25Q8×M16C2212yes—
26C2×C4×Q16C2211yescharacteristic
27C2×C8×D4C2211yescharacteristic
28C22×#103C412yes—
29#1998—22no—
30#1997—22no—
31#1997C2212yes—
32#1997C2212yes—
33#1800—21no—
34#1800—21no—
35#1800C2211yescharacteristic
36#1800C2211yescharacteristic
37C4×#100—21no—
38C4×#100—21no—
39C4×#100C2211yescharacteristic
40#1593—22no—
41#1587—22no—
42#1587—22no—
43#1579—22no—
44#1566C2212yes—
45#1549C2212yes—
46#1548C2212yes—
47#1487C2212yes—
48#1487C2212yes—
49#1460C2211yescharacteristic
50#1451C2212yes—
51#1292C2212yes—
52#1283C2212yes—
53C23×Q8C2×C412yes—
54C22×Q16C2×C412yes—
55C22×Q16C2×C411yescharacteristic
56C23×C8C2×C412yes—
57#554—22no—
58#554—22no—
59#554—42no—
60D4×Q8—22no—
61D4×Q8—22no—
62D4×Q8—42no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×#125—21no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8C2×C412yes—
70C2×C22⋊Q8C2×C412yes—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C22×C4⋊C4C2×C412yes—
77#487—22no—
78#487—22no—
79#484—24no—
80#484—24no—
81#477—22no—
82#477—22no—
83#467—24no—
84#467—24no—
85#462—22no—
86#462—22no—
87#451—24no—
88#451—24no—
89#446—22no—
90#446—22no—
91#446—42no—
92#446—42no—
93C8×Q8—22no—
94C8×Q8—22no—
95C4×Q16D412yes—
96C4×Q16D412yes—
97C4×Q16D412yes—
98C4×Q16D412yes—
99C8×D4—22no—
100C8×D4—22no—
101#431C2312yes—
102C2×#107C2×C412yes—
103C2×#107C2×C411yescharacteristic
104#423—22no—
105#423—22no—
106#423C2312yes—
107#423C2312yes—
108C2×C4⋊C8C2311yescharacteristic
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103C2×C412yes—
114C2×#103C2×C412yes—
115C2×#103C2×C412yes—
116C2×#103C2×C412yes—
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#100—21no—
120C2×#100—21no—
121C2×#100D411yescharacteristic
122C2×#100D411yescharacteristic
123C2×C22⋊C8C2×C412yes—
124C4×M16—22no—
125C4×M16—22no—
126C4×M16C2312yes—
127C2×C4×C8C2311yescharacteristic
128#343—22no—
129#343—22no—
130#343—22no—
131#343—22no—
132#343—21no—
133#343—21no—
134#343D412yes—
135#334—22no—
136#334—22no—
137#333—42no—
138#332—42no—
139Q8⋊C8—22no—
140Q8⋊C8—22no—
141Q8⋊C8—22no—
142Q8⋊C8—22no—
143#323—22no—
144#322—22no—
145#322—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—42no—
151C22×Q8—44no—
152C22×Q8—82no—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C22×C412yes—
155C22×Q8C22×C411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C22×C412yes—
160C2×Q16—24no—
161C2×Q16—42no—
162C2×Q16—82no—
163C2×Q16C22⋊C412yes—
164C2×Q16C22⋊C412yes—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16C2×D412yes—
168C2×M16C2×D412yes—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—82no—
172C22×C8C22×C412yes—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181#125—42no—
182#125—42no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8C2×D412yes—
194C4×Q8C2×D412yes—
195C4×Q8C2×D412yes—
196C4×Q8C2×D412yes—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4#4112yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C2411yescharacteristic
224#107—24no—
225#107C22⋊C412yes—
226#107C22⋊C412yes—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237#103—22no—
238#103—22no—
239#103—22no—
240#103—22no—
241#103—24no—
242#103—22no—
243#103—22no—
244#103C22⋊C411yescharacteristic
245#103C22⋊C411yescharacteristic
246#103C22⋊C411yescharacteristic
247#103C22⋊C411yescharacteristic
248#103C22⋊C412yes—
249#103C22⋊C412yes—
250#100—22no—
251#100—22no—
252#100—42no—
253#100—42no—
254C22⋊C8—24no—
255C22⋊C8—22no—
256#97—22no—
257#97—42no—
258#97—42no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8C2×D412yes—
264C4×C8C2×D412yes—
265C24C4×D412yes—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—84no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—88no—
285C2×Q8—161no—
286C2×Q8C2×C22⋊C411yescharacteristic
287C2×Q8C2×C22⋊C412yes—
288C2×Q8C2×C22⋊C411yescharacteristic
289C2×D4—22no—
290C2×D4—22no—
291C2×D4#13712yes—
292C2×D4#13712yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—84no—
308C22×C4—162no—
309C22×C4C4×D412yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313Q16—24no—
314Q16—42no—
315Q16—84no—
316M16—22no—
317M16—22no—
318M16—22no—
319M16—22no—
320M16—42no—
321M16—42no—
322M16—42no—
323M16—42no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—82no—
334C2×C8—82no—
335C2×C8C2×C22⋊C412yes—
336C2×C8C2×C22⋊C411yescharacteristic
337C2×C8C2×C22⋊C411yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4C2×C22⋊C411yescharacteristic
355C4⋊C4C2×C22⋊C411yescharacteristic
356C4⋊C4C2×C22⋊C412yes—
357C22⋊C4—24no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42C22×D412yes—
365C42H(5,𝔽2)12yes—
366C23—22no—
367C23—22no—
368C23—42no—
369C23—42no—
370C23#44114yes—
371C23C2×C4×D411yescharacteristic
372Q8—21no—
373Q8—22no—
374Q8—21no—
375Q8—22no—
376Q8—21no—
377Q8—42no—
378Q8—42no—
379Q8—42no—
380Q8—81no—
381Q8—84no—
382Q8—82no—
383Q8—84no—
384Q8—162no—
385Q8#41312yes—
386D4—24no—
387D4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—162no—
413C2×C4—162no—
414C2×C4C22×C22⋊C411yescharacteristic
415C2×C4#51811yescharacteristic
416C2×C4D4212yes—
417C2×C4#54612yes—
418C2×C4C2×#13711yescharacteristic
419C2×C4C2×#13711yescharacteristic
420C2×C4#57812yes—
421C8—22no—
422C8—21no—
423C8—21no—
424C8—42no—
425C8—42no—
426C8—42no—
427C8—82no—
428C8—82no—
429C8—82no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—44no—
434C22—44no—
435C22—42no—
436C22—42no—
437C22D4×C22⋊C411yescharacteristic
438C22C2×#44111yescharacteristic
439C22#295011yescharacteristic
440C4—21no—
441C4—22no—
442C4—22no—
443C4—42no—
444C4—41no—
445C4—42no—
446C4—42no—
447C4—41no—
448C4—41no—
449C4—82no—
450C4—82no—
451C4C2×#41311yescharacteristic
452C4#288811yescharacteristic
453C4#327012yes—
454C4#327012yes—
455C2—21no—
456C2—21no—
457C2—44no—
458C2#2240811yescharacteristic
459C2#2244011yescharacteristic
460C2#2245911yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #18042C2semidirect
5. #9803C2semidirect
6. #9250C2semidirect
7. #9248C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32992
#225152
Q8×#1002
#180422
#98032
#92502
#92482
#86142
#83952
#83912
#81642