Finite Group Databasebeta

← #112816

Subgroups — #112816

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#17742C211yesmaximal, characteristic
6#7885C211yesmaximal, characteristic
7#7885C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10C2×#498—22no—
11C2×#498—22no—
12C2×#498—22no—
13C2×#498—22no—
14C2×#498—22no—
15C2×#498C414yes—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C22×#106—22no—
21C22×C4×C8C2211yescharacteristic
22#1997—22no—
23#1997—22no—
24#1821—22no—
25#1558—22no—
26#1556—22no—
27#1556—22no—
28#1549—22no—
29#1549—22no—
30#1548—22no—
31#1548—22no—
32#1546—22no—
33#1546—22no—
34#1278—22no—
35#1278—22no—
36C23×Q8—21no—
37C23×Q8—21no—
38C23×C8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8D412yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C42C2311yescharacteristic
68#498—28no—
69#498—28no—
70#498—28no—
71#498—28no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88C2×#106—28no—
89C2×#106—28no—
90C2×#106—28no—
91#423D412yes—
92#423D412yes—
93#423D412yes—
94#423D412yes—
95#407—21no—
96#407—21no—
97#407—21no—
98#407—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2×C412yes—
107C2×C4×C8D412yes—
108C2×C4×C8D412yes—
109#343—28no—
110#333—28no—
111#332—28no—
112#332—28no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—41no—
119C22×Q8—41no—
120C22×Q8—41no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—41no—
124C22×Q8—41no—
125C22×Q8—84no—
126C22×Q8—84no—
127C23×C4—22no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C22×C8—28no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8C22⋊C414yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C22⋊C414yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208#106—232no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C22⋊C8—41no—
221C22⋊C8—41no—
222C22⋊C8—41no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225#97—41no—
226#97—41no—
227#97—41no—
228#97—41no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—41no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8—41no—
236C4×C8C22⋊C414yes—
237C24#12311yescharacteristic
238C2×Q8—24no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×Q8—84no—
249C2×Q8—84no—
250C2×Q8—84no—
251C2×Q8—84no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—84no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12411yescharacteristic
286C22×C4#12411yescharacteristic
287M16—44no—
288M16—44no—
289M16—44no—
290M16—44no—
291C2×C8—28no—
292C2×C8—28no—
293C2×C8—28no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C42C2×C22⋊C411yescharacteristic
316C42C2×C22⋊C411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42#12011yescharacteristic
320C42#12011yescharacteristic
321C42#12011yescharacteristic
322C42#12011yescharacteristic
323C42#12011yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42#12311yescharacteristic
329C42#12711yescharacteristic
330C42#12711yescharacteristic
331C23—22no—
332C23—22no—
333C23—22no—
334C23#38611yescharacteristic
335C23#39411yescharacteristic
336C23#39411yescharacteristic
337Q8—84no—
338Q8—84no—
339Q8—84no—
340Q8—84no—
341Q8—84no—
342Q8—84no—
343Q8—84no—
344Q8—84no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4#37911yescharacteristic
369C2×C4#38611yescharacteristic
370C2×C4#38611yescharacteristic
371C2×C4#38611yescharacteristic
372C2×C4#39011yescharacteristic
373C2×C4#39011yescharacteristic
374C2×C4#39211yescharacteristic
375C2×C4#39211yescharacteristic
376C2×C4#39211yescharacteristic
377C2×C4#39211yescharacteristic
378C2×C4#39411yescharacteristic
379C2×C4#39411yescharacteristic
380C2×C4#45614yes—
381C2×C4#47114yes—
382C2×C4#47114yes—
383C8—28no—
384C8—82no—
385C8—82no—
386C8—82no—
387C8—82no—
388C22—28no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22#143811yescharacteristic
396C22#188612yes—
397C22#204612yes—
398C22#204612yes—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—84no—
406C4—84no—
407C4#197712yes—
408C4#197712yes—
409C4#200512yes—
410C4#200512yes—
411C4#201912yes—
412C4#201912yes—
413C2—22no—
414C2—22no—
415C2—22no—
416C2#1097811yescharacteristic
417C2#1102811yescharacteristic
418C2#1102811yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#4982
#218292
#177462
#177422
#78852