Finite Group Databasebeta

← #193373

Subgroups — #193373

Order 512.

793 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21970C211yesmaximal, characteristic
3#17816C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5#17501C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7883C211yesmaximal, characteristic
9#7876C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962C2214yes—
16#2961—22no—
17#2924—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#1950—22no—
25#1950C2214yes—
26#1811—22no—
27#1572—22no—
28#1571C2212yes—
29#1570C2212yes—
30#1568—22no—
31#1550C414yes—
32#1549C412yes—
33#1548C412yes—
34#1542C2212yes—
35#1542C2212yes—
36#1493C2214yes—
37#1491—22no—
38#1485—22no—
39#1278C2212yes—
40#1278C2212yes—
41#1277C2214yes—
42C22×M16—21no—
43C22×M16—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50#533Q812yes—
51#533C2312yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8Q812yes—
57C2×C4×Q8—21no—
58C2×C4×Q8D412yes—
59C2×#117—21no—
60C22×C42C2311yescharacteristic
61#446—28no—
62#446—28no—
63#446—28no—
64#446—24no—
65C8×Q8—24no—
66#433—28no—
67#432—24no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432D412yes—
75#432D412yes—
76#431—24no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431C2×C412yes—
84#431C2311yescharacteristic
85#431C2311yescharacteristic
86C2×#108—24no—
87#424—24no—
88#423—21no—
89#423—21no—
90#423C2×C412yes—
91#423C2×C412yes—
92#423D412yes—
93#423C2312yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C412yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98#407—21no—
99#407D412yes—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102#405—24no—
103C4×M16—21no—
104C4×M16—21no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16—21no—
108C4×M16C2312yes—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2312yes—
113#341—24no—
114#335—24no—
115#334—24no—
116#333C2×C414yes—
117#333C2×C414yes—
118#332C2×C414yes—
119#332C2×C414yes—
120Q8⋊C8—28no—
121C22×Q8—21no—
122C22×Q8C4⋊C412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q811yescharacteristic
125C23×C4#4111yescharacteristic
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—44no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—82no—
133C2×M16C2×D414yes—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C4⋊C412yes—
143C4⋊Q8C4⋊C412yes—
144C4⋊Q8C22×C412yes—
145C4⋊Q8C22×C412yes—
146#125—22no—
147#125—22no—
148#125—22no—
149#125—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8C4⋊C414yes—
157C22⋊Q8C2×Q814yes—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8C4⋊C414yes—
165C4×Q8C2×D414yes—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C4⋊C412yes—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42C2411yescharacteristic
190#108—84no—
191C4⋊C8—28no—
192C4⋊C8—28no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8C22×C412yes—
203C4⋊C8C22×C412yes—
204C4⋊C8C22×C414yes—
205C4⋊C8C2×D414yes—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#97—28no—
216#97—22no—
217#97—22no—
218#97—22no—
219#97—41no—
220#97—41no—
221#97—41no—
222C4×C8—28no—
223C4×C8—22no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—22no—
229C4×C8—41no—
230C24C22⋊Q811yescharacteristic
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8C2×C4⋊C414yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4C2×C4⋊C411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×Q811yescharacteristic
260C22×C4C2×#4111yescharacteristic
261M16—24no—
262M16—24no—
263M16—44no—
264M16—44no—
265M16—42no—
266M16—42no—
267M16—82no—
268M16—82no—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—82no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4C2×C4⋊C414yes—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C42—24no—
297C42—24no—
298C42C2×C4⋊C411yescharacteristic
299C42C2×C4⋊C411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42C4×D411yescharacteristic
304C42#12011yescharacteristic
305C42#12011yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42#12311yescharacteristic
309C42#12311yescharacteristic
310C42C23×C411yescharacteristic
311C42C22×D411yescharacteristic
312C42C22×D411yescharacteristic
313C42C2×#4111yescharacteristic
314C23—22no—
315C23—41no—
316C23—41no—
317C23#38511yescharacteristic
318C23#38511yescharacteristic
319C23C2×C22⋊Q811yescharacteristic
320Q8—44no—
321Q8—44no—
322Q8—82no—
323Q8—82no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4#38511yescharacteristic
350C2×C4#38511yescharacteristic
351C2×C4#38511yescharacteristic
352C2×C4#38511yescharacteristic
353C2×C4C22×C4⋊C411yescharacteristic
354C2×C4C2×C4×D411yescharacteristic
355C2×C4C2×C4×D411yescharacteristic
356C2×C4C2×#12011yescharacteristic
357C2×C4C2×C22⋊Q811yescharacteristic
358C2×C4C2×#12311yescharacteristic
359C2×C4#57712yes—
360C2×C4#57812yes—
361C8—44no—
362C8—44no—
363C8—42no—
364C8—42no—
365C8—44no—
366C8—44no—
367C8—82no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—82no—
377C22C2×#38511yescharacteristic
378C22#308212yes—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—44no—
389C4—44no—
390C4#294911yescharacteristic
391C4#295011yescharacteristic
392C4#295612yes—
393C4#301312yes—
394C2—22no—
395C2—41no—
396C2—41no—
397C2#2086011yescharacteristic
398C2#2088011yescharacteristic
399C2#2088511yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8933C2semidirect
7. #8931C2semidirect

Maximal subgroups

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

subgroupindex
#602512
#219702
#178162
#177082
#175012
#89332
#89312
#78832
#78762