Finite Group Databasebeta

← #198699

Subgroups — #198699

Order 512.

1037 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#60251C211yesmaximal, characteristic
3#21829C211yesmaximal, characteristic
4#17748C211yesmaximal, characteristic
5C2×#1558C211yesmaximal, characteristic
6#9268C218yesmaximal, complemented
7#8080C211yesmaximal, characteristic
8#7933C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—22no—
12#3209—22no—
13#3209C414yes—
14#3209C2214yes—
15C2×#494—21no—
16C2×#487—21no—
17C2×#462—22no—
18C2×#451C414yes—
19#2987—22no—
20#2985—22no—
21#2984—22no—
22#2983—22no—
23#2962C2214yes—
24#2961—22no—
25#2924—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#1997—22no—
32#1997C2214yes—
33#1821—22no—
34#1558—22no—
35#1558—22no—
36#1558—21no—
37#1556—21no—
38#1554—22no—
39#1550C2214yes—
40C2×#333C2214yes—
41#1301C2214yes—
42#1283C2214yes—
43C23×Q8C2×C412yes—
44C22×Q16—24no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
50#533D412yes—
51#533D412yes—
52#533C2312yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8C2312yes—
67C2×#125—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8C2×C412yes—
70C2×C4×Q8—21no—
71C2×C4×Q8D412yes—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#494—24no—
77#494—24no—
78#487—24no—
79#487—24no—
80#462—28no—
81#462—28no—
82#451—28no—
83#451C2×C4116yes—
84#446—28no—
85#446—28no—
86#446—28no—
87#446—24no—
88C8×Q8—24no—
89#433—28no—
90#432—24no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432D412yes—
102#432D412yes—
103#431—24no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2×C412yes—
111#431C2312yes—
112#424—24no—
113#423D412yes—
114#423C2312yes—
115C2×C4⋊C8—21no—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×#103—24no—
119#407—21no—
120#407D412yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8C2×C412yes—
124#405—24no—
125C4×M16—21no—
126C4×M16C2312yes—
127C2×#97—21no—
128C2×#97—21no—
129C2×C4×C8—21no—
130#343—28no—
131#333—24no—
132#333—28no—
133#332—24no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—84no—
141C22×Q8C22⋊C412yes—
142C22×Q8C22×C414yes—
143C23×C4—22no—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4#4111yescharacteristic
147C2×Q16—216no—
148C2×Q16—88no—
149C2×M16—22no—
150C2×M16C2×D414yes—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8C22⋊C412yes—
175C4⋊Q8C22⋊C412yes—
176C4⋊Q8C2×D412yes—
177C4⋊Q8C2×D412yes—
178#125—22no—
179#125—22no—
180#125—22no—
181#125—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8C22⋊C414yes—
189C22⋊Q8C22×C414yes—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8C22⋊C414yes—
197C4×Q8C2×D414yes—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C22⋊C412yes—
220C2×C22⋊C4—22no—
221C2×C42—22no—
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42C2411yescharacteristic
234C4⋊C8—28no—
235C4⋊C8—28no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8C2×D414yes—
246#103—216no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—41no—
253C22⋊C8—41no—
254C22⋊C8—41no—
255C22⋊C8—41no—
256C22⋊C8C22×C414yes—
257#97—28no—
258#97—22no—
259#97—22no—
260#97—22no—
261#97—41no—
262#97—41no—
263#97—41no—
264C4×C8—28no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—22no—
268C4×C8—41no—
269C24C4×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—44no—
276C2×Q8—48no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—82no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8C2×C22⋊C414yes—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—84no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313Q16—816no—
314M16—28no—
315M16—44no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—82no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4C2×C22⋊C414yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C42—24no—
347C42—24no—
348C42C2×C22⋊C411yescharacteristic
349C42C2×C22⋊C411yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42#12011yescharacteristic
353C42#12011yescharacteristic
354C42#12111yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42#12111yescharacteristic
358C42#12311yescharacteristic
359C42#12311yescharacteristic
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C22×D411yescharacteristic
363C42C2×#4111yescharacteristic
364C23—22no—
365C23—22no—
366C23—41no—
367C23#38611yescharacteristic
368C23#38611yescharacteristic
369C23C2×C4×D411yescharacteristic
370Q8—44no—
371Q8—48no—
372Q8—44no—
373Q8—82no—
374Q8—82no—
375Q8—84no—
376Q8—84no—
377Q8—84no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4#38611yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4#38611yescharacteristic
406C2×C4#38611yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38611yescharacteristic
409C2×C4C22×C22⋊C411yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C2×#12011yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12311yescharacteristic
415C2×C4#57814yes—
416C8—44no—
417C8—44no—
418C8—44no—
419C8—82no—
420C8—82no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—44no—
429C22#191412yes—
430C22C2×#38611yescharacteristic
431C22#295012yes—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—44no—
442C4—44no—
443C4—84no—
444C4#301112yes—
445C4#304412yes—
446C4#306912yes—
447C2—22no—
448C2—22no—
449C2—41no—
450C2#2093511yescharacteristic
451C2#2100611yescharacteristic
452C2C2×#191411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
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. #9268C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#602512
#218292
#177482
C2×#15582
#92682
#80802
#79332