Finite Group Databasebeta

← #224190

Subgroups — #224190

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#124C211yesmaximal, characteristic
2#21346C212yesmaximal, complemented
3#9895C212yesmaximal, complemented
4#9895C212yesmaximal, complemented
5#9890C212yesmaximal, complemented
6#8583C211yesmaximal, characteristic
7#8581C212yesmaximal
8#8226C212yesmaximal
9#8224C211yesmaximal, characteristic
10C2×Q82C411yescharacteristic
11C2×D4×Q8C411yescharacteristic
12#2862C2212yes—
13#2862C2211yescharacteristic
14#2568C412yes—
15#2568C411yescharacteristic
16#2563C411yescharacteristic
17#2563C412yes—
18#2492—22no—
19#2492—22no—
20#2492—22no—
21Q8×C22⋊C4—22no—
22C4×#124C2211yescharacteristic
23C4×#124C2212yes—
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1940—22no—
27#1916—22no—
28#1916—22no—
29#1767C2212yes—
30#1767C2212yes—
31#1767C2212yes—
32#1745C2212yes—
33#1745C2212yes—
34C4×#97C2212yes—
35C2×#329C2211yescharacteristic
36C2×#329C2212yes—
37C2×#329C2211yescharacteristic
38#1295C2212yes—
39#1295C2212yes—
40#1295C2212yes—
41#1295C2212yes—
42#1295C2212yes—
43#1295C2211yescharacteristic
44#1295C2211yescharacteristic
45C23×Q8—21no—
46Q82—44no—
47D4×Q8—44no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C412yes—
54C2×#124C2311yescharacteristic
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C412yes—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×#97—22no—
68C2×#97C2312yes—
69C2×#97C2312yes—
70C2×#97C2312yes—
71C2×#97C2312yes—
72C2×C4×C8—22no—
73#400—21no—
74#400—21no—
75#400—22no—
76#400—22no—
77#393—21no—
78#393—21no—
79#393—22no—
80#393—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391C2×C411yescharacteristic
91#391C2×C412yes—
92#390C2×C411yescharacteristic
93#390C2×C412yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#382C2×C412yes—
98#382C2×C412yes—
99#382C2×C412yes—
100#382C2311yescharacteristic
101#382C2312yes—
102#380—22no—
103#380—22no—
104#380—22no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C43C2312yes—
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#329—22no—
123#329—24no—
124#329Q814yescomplemented
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—44no—
130C22×Q8—48no—
131C22×Q8C22⋊C411yescharacteristic
132C22×Q8C22×C411yescharacteristic
133C22×D4C22×C411yescharacteristic
134C23×C4—22no—
135C23×C4—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C4⋊Q8—42no—
141C4⋊Q8—44no—
142C4⋊Q8—44no—
143C4⋊Q8—42no—
144C4⋊Q8—48no—
145C4⋊Q8—44no—
146#124—22no—
147#124C2×Q812yes—
148#124#4112yes—
149C22⋊Q8—48no—
150C22⋊Q8—44no—
151C4×Q8—24no—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—42no—
155C4×Q8—44no—
156C4×D4—42no—
157C4×D4—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2411yescharacteristic
192C4⋊C8—44no—
193C4⋊C8—44no—
194C4⋊C8—44no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—44no—
203#97C2×Q814yescomplemented
204C4×C8—44no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95C22⋊C411yescharacteristic
218#95C22⋊C411yescharacteristic
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221C24—21no—
222C2×Q8—21no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×Q8—816no—
238C2×Q8C4×Q812yes—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4C4×Q812yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D412yes—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4#10014yes—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—24no—
297C42—22no—
298C42—24no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—41no—
304C42—42no—
305C42—42no—
306C42—44no—
307C42—42no—
308C42C22×Q811yescharacteristic
309C42C2×#4111yescharacteristic
310C422−1+412yes—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23C22×C22⋊C411yescharacteristic
317Q8—28no—
318Q8—42no—
319Q8—42no—
320Q8—84no—
321Q8—88no—
322Q8—164no—
323Q8—162no—
324D4—42no—
325D4—42no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—84no—
360C2×C4—88no—
361C2×C4C2×#10012yes—
362C2×C4C2×#10014yes—
363C2×C4#42114yes—
364C2×C4C2×C4×Q811yescharacteristic
365C2×C4#51911yescharacteristic
366C2×C4#54812yes—
367C2×C4D4×Q812yes—
368C8—44no—
369C8—44no—
370C8—82no—
371C8—82no—
372C22—44no—
373C22—42no—
374C22—81no—
375C22—82no—
376C22—81no—
377C22Q8×C22⋊C411yescharacteristic
378C22C22×#10011yescharacteristic
379C22#288911yescharacteristic
380C22C2×#42112yes—
381C22#290112yes—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—24no—
386C4—24no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—41no—
393C4—42no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—82no—
398C2—82no—
399C2#1766611yescharacteristic
400C2#1768312yes—
401C2#1768511yescharacteristic
402C2Q8×#10011yescharacteristic
403C2#2252912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21346C2semidirect
3. #9895C2semidirect
4. #9895C2semidirect
5. #9890C2semidirect
124. #329Q8semidirect
203. #97C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1242
#213462
#98952
#98902
#85832
#85812
#82262
#82242