Finite Group Databasebeta

← #189045

Subgroups — #189045

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#17817C211yesmaximal, characteristic
3#17814C211yesmaximal, characteristic
4#17751C211yesmaximal, characteristic
5C2×#1561C211yesmaximal, characteristic
6#8641C214yesmaximal, complemented
7#8639C214yesmaximal, complemented
8#8129C211yesmaximal, characteristic
9#8128C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2986—22no—
12#2985—22no—
13#2984—22no—
14#2983—21no—
15#2982—21no—
16#2962C414yes—
17#2960C2214yes—
18#2924C414yes—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25#1574—22no—
26#1573—22no—
27#1571—21no—
28#1571—21no—
29#1571—22no—
30#1567C2214yes—
31#1567C2212yes—
32#1566C2212yes—
33#1561—22no—
34#1561C2214yes—
35#1561C2212yes—
36#1561C2212yes—
37#1552—22no—
38#1549—21no—
39#1546—21no—
40C2×#332—21no—
41C2×#332—21no—
42#1309C2212yes—
43#1309C2214yes—
44#1309C2212yes—
45C22×M16—21no—
46C22×M16—21no—
47C22×M16—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533D412yes—
54#533Q812yes—
55#533C2312yes—
56C2×C4⋊Q8D411yescharacteristic
57C2×C4⋊Q8D411yescharacteristic
58C2×#125Q812yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×#117C2×C412yes—
64C22×C42C2311yescharacteristic
65#446—24no—
66#446—28no—
67#446—24no—
68#446—24no—
69C8×Q8—24no—
70C8×Q8—24no—
71C8×Q8—24no—
72#433—28no—
73#432—21no—
74#432—21no—
75#432—24no—
76#432—21no—
77#432—21no—
78#431—21no—
79#431—21no—
80#431—24no—
81#431—21no—
82#431—21no—
83#431D411yescharacteristic
84#431D411yescharacteristic
85#431D412yes—
86#431D412yes—
87#431D412yes—
88#424C2×C418yes—
89#423—21no—
90#423—21no—
91#423—21no—
92#423—21no—
93#423—21no—
94#423—21no—
95#423—21no—
96#423—21no—
97#423—21no—
98#423C2311yescharacteristic
99#423C2312yes—
100#423C2312yes—
101#423C2312yes—
102#423C2311yescharacteristic
103#423C2312yes—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2×C412yes—
111#407—21no—
112#407C2×C412yes—
113C2×C22⋊C8C2×C412yes—
114#405C2×C418yes—
115C4×M16—21no—
116C4×M16C2×C412yes—
117C2×C4×C8C2312yes—
118#335—24no—
119#335—24no—
120#335—24no—
121#334—24no—
122#332—24no—
123#332—24no—
124#332—22no—
125#332—22no—
126#332—22no—
127#332—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C23×C4C22×C411yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—41no—
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—82no—
142C2×M16—82no—
143C2×M16—82no—
144C2×M16C22×C414yes—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8—41no—
148C22×C8C22×C414yes—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C4⋊C412yes—
154C4⋊Q8C4⋊C412yes—
155C4⋊Q8C2×D412yes—
156C4⋊Q8C2×D412yes—
157#125—22no—
158#125—22no—
159#125C4⋊C414yes—
160#125C2×Q814yes—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177#117—22no—
178#117—22no—
179#117C4⋊C414yes—
180#117C22×C414yes—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C4⋊C412yes—
186C2×C4⋊C4C4⋊C412yes—
187C2×C22⋊C4—22no—
188C2×C42—22no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×Q811yescharacteristic
197C2×C42C2×Q811yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201C4⋊C8—28no—
202C4⋊C8—28no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—21no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8C22⋊C412yes—
222C4⋊C8C22⋊C412yes—
223C4⋊C8C22⋊C414yes—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8C22⋊C412yes—
229C22⋊C8C22⋊C412yes—
230C22⋊C8C22⋊C414yes—
231#97—28no—
232#97—22no—
233#97—41no—
234C4×C8—28no—
235C4×C8—22no—
236C4×C8—41no—
237C4×C8C2×D414yes—
238C4×C8C2×D412yes—
239C4×C8C2×D412yes—
240C24#11711yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—81no—
248C2×Q8—81no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4#11711yescharacteristic
261C22×C4#11711yescharacteristic
262C22×C4#11711yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274M16—44no—
275M16—82no—
276M16—82no—
277M16—82no—
278M16—81no—
279M16—81no—
280M16—82no—
281M16—82no—
282M16—82no—
283C2×C8—24no—
284C2×C8—44no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—82no—
291C2×C8—82no—
292C2×C8—82no—
293C2×C8C2×C22⋊C414yes—
294C2×C8C2×C22⋊C414yes—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4C2×C4⋊C414yes—
306C4⋊C4C2×C4⋊C414yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C42—24no—
310C42—24no—
311C42C2×C22⋊C411yescharacteristic
312C42C2×C22⋊C411yescharacteristic
313C42C2×C4⋊C411yescharacteristic
314C42C2×C4⋊C411yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42#12111yescharacteristic
318C42#12111yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42C22×D411yescharacteristic
324C42C22×D411yescharacteristic
325C42C22×D411yescharacteristic
326C42C22×Q811yescharacteristic
327C23—22no—
328C23—41no—
329C23—41no—
330C23#38011yescharacteristic
331C23#38011yescharacteristic
332C23C2×#11711yescharacteristic
333Q8—44no—
334Q8—42no—
335Q8—42no—
336Q8—42no—
337Q8—42no—
338Q8—82no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4#38011yescharacteristic
366C2×C4#38011yescharacteristic
367C2×C4#38011yescharacteristic
368C2×C4#38011yescharacteristic
369C2×C4#38011yescharacteristic
370C2×C4#38011yescharacteristic
371C2×C4#47212yes—
372C2×C4#47312yes—
373C2×C4C22×C22⋊C411yescharacteristic
374C2×C4C22×C4⋊C411yescharacteristic
375C2×C4C2×#12111yescharacteristic
376C2×C4C2×#12111yescharacteristic
377C2×C4C2×C22⋊Q811yescharacteristic
378C2×C4C2×C22⋊Q811yescharacteristic
379C8—24no—
380C8—24no—
381C8—44no—
382C8—82no—
383C8—82no—
384C8—82no—
385C8—81no—
386C8—81no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—82no—
396C22#181612yes—
397C22C2×#38011yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—44no—
408C4#181612yes—
409C4#288912yes—
410C4C2×#47211yescharacteristic
411C4C2×#47311yescharacteristic
412C4#306811yescharacteristic
413C4#306911yescharacteristic
414C2—22no—
415C2—41no—
416C2—41no—
417C2#2063911yescharacteristic
418C2C2×#181611yescharacteristic
419C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8641C2semidirect
7. #8639C2semidirect

Maximal subgroups

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

subgroupindex
#602502
#178172
#178142
#177512
C2×#15612
#86412
#86392
#81292
#81282