Finite Group Databasebeta

← #138336

Subgroups — #138336

Order 512.

710 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1820C211yesmaximal, characteristic
2#14887C211yesmaximal, characteristic
3C2×#1351C212yesmaximal
4#8300C211yesmaximal, characteristic
5#7250C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#395C411yescharacteristic, complemented
8C2×#393C2211yescharacteristic
9C2×#391C411yescharacteristic, complemented
10C2×#381—21no—
11#1820—24no—
12#1817—21no—
13C2×#342—21no—
14C2×#342C2212yes—
15C2×#324—21no—
16C2×#323C2212yes—
17#1442—21no—
18#1441C411yescharacteristic, complemented
19#1439—21no—
20#1439C2211yescharacteristic
21#1436—21no—
22#1434C411yescharacteristic, complemented
23C22⋊#95—21no—
24#1351—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—42no—
30C2×#117—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C22⋊C4—22no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D411yescharacteristic
39C22×C22⋊C4C2311yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42D411yescharacteristic
42C2×C22⋊C8—22no—
43#395—42no—
44#393—22no—
45#393—22no—
46#391—42no—
47#381—24no—
48#381—42no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—42no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61#342—24no—
62#342—22no—
63#342D414yes—
64#336—42no—
65#324—42no—
66#323—22no—
67#323—22no—
68C25C2×D411yescharacteristic
69C22×Q8—22no—
70C22×Q8—42no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—81no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—42no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—42no—
94#117—48no—
95#117—48no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4—81no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—81no—
125C2×C22⋊C4—81no—
126C2×C22⋊C4C2×D412yes—
127C2×C22⋊C4#4112yes—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—41no—
133C2×C42—41no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136#95—22no—
137#95—41no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—48no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—81no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24C2×C22⋊C411yescharacteristic
157C24#12011yescharacteristic
158C24#12311yescharacteristic
159C24#13512yes—
160C24#13712yes—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—82no—
164C2×Q8—84no—
165C2×Q8—84no—
166C2×Q8—82no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4#9912yes—
243C22×C4#9912yes—
244C22×C4C4×D412yes—
245C22×C4#12112yes—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—84no—
252C4⋊C4—82no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—84no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C42—44no—
265C42—44no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—41no—
282C23—41no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23#38611yescharacteristic
297C23C2×#9911yescharacteristic
298C23C2×#9911yescharacteristic
299C23#41611yescharacteristic
300C23#41911yescharacteristic
301C23#42111yescharacteristic
302C23#42111yescharacteristic
303C23#45212yes—
304C23#48512yes—
305Q8—84no—
306Q8—84no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—84no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—162no—
375C2×C4—162no—
376C2×C4—162no—
377C2×C4—162no—
378C8—162no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—81no—
405C22—81no—
406C22—82no—
407C22#151111yescharacteristic
408C22#151111yescharacteristic
409C22#151611yescharacteristic
410C22#151611yescharacteristic
411C22#187311yescharacteristic
412C22#189711yescharacteristic
413C22#189811yescharacteristic
414C22#212011yescharacteristic
415C22#212211yescharacteristic
416C22#212611yescharacteristic
417C22#212911yescharacteristic
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—82no—
423C4—81no—
424C4—82no—
425C4—81no—
426C4—161no—
427C4—164no—
428C4—161no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—42no—
435C2#915411yescharacteristic
436C2#1181011yescharacteristic
437C2#1181211yescharacteristic
438C2#1184211yescharacteristic
439C2#1184411yescharacteristic
440C2#1308711yescharacteristic
441C2#1309111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#395C4semidirect
9. C2×#391C4semidirect
18. #1441C4semidirect
22. #1434C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18202
#148872
C2×#13512
#83002
#72502