Finite Group Databasebeta

← #46839990

Subgroups — #46839990

Order 34688.

836 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16864654C211yesmaximal, characteristic
2C2×#22576159C211yesmaximal, characteristic, complemented
3C2×#22576146C211yesmaximal, characteristic, complemented
4C2×#22576138C211yesmaximal, characteristic, complemented
5#31773815C211yesmaximal, characteristic, complemented
6#31773814C211yesmaximal, characteristic, complemented
7#31773813C211yesmaximal, characteristic, complemented
8#31773812C211yesmaximal, characteristic, complemented
9#31773801C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773771C211yesmaximal, characteristic, complemented
12#31773770C211yesmaximal, characteristic, complemented
13#31773768C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic, complemented
15#385×C271C211yesmaximal, characteristic, complemented
16C2×#16864654—24no—
17C2×#16864654—24no—
18C2×#16864654—24no—
19C22×Dic542C2211yescharacteristic
20C22×Dic542C2211yescharacteristic
21C22×D1084C2211yescharacteristic
22C22×C22⋊D271C2211yescharacteristic
23C22×C22⋊D271C2211yescharacteristic
24C22×C4×D271C2211yescharacteristic
25C22×C4×D271C2211yescharacteristic
26#22576159—24no—
27#22576146—24no—
28#22576138—24no—
29C2×C4⋊Dic271C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×Dic271⋊C4C2211yescharacteristic
32C2×Dic271⋊C4C2211yescharacteristic
33C2×Dic271⋊C4C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic
43C2×(C2×D271)⋊C4C2211yescharacteristic
44C2×C4×Dic271C2211yescharacteristic
45C2×C4×Dic271C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C271⋊#95C2211yescharacteristic
49C271⋊#95C2211yescharacteristic
50C23×C4×C271C2211yescharacteristic
51C2×C4⋊C4×C271C2211yescharacteristic
52C2×C4⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55#95×C271C2211yescharacteristic
56#95×C271C2211yescharacteristic
57C23×D271C2311yescharacteristic
58C23×D271C2311yescharacteristic
59#16864654—232no—
60C2×Dic542—22no—
61C2×Dic542—22no—
62C2×Dic542—22no—
63C2×Dic542—22no—
64C2×Dic542—22no—
65C2×Dic542D414yes—
66C2×D1084—22no—
67C2×D1084—22no—
68C2×D1084—22no—
69C2×C22⋊D271—22no—
70C2×C22⋊D271—22no—
71C2×C22⋊D271—22no—
72C2×C22⋊D271—22no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271D414yes—
75C22×Dic271C2311yescharacteristic
76C22×Dic271C2311yescharacteristic
77C22×Dic271C2311yescharacteristic
78C22×Dic271C2311yescharacteristic
79C22×Dic271C2311yescharacteristic
80C22×Dic271C2311yescharacteristic
81C2×C4×D271—22no—
82C2×C4×D271—22no—
83C2×C4×D271—22no—
84C2×C4×D271—22no—
85C2×C4×D271—22no—
86C2×C4×D271D414yes—
87C4⋊Dic271—22no—
88Dic271⋊C4—22no—
89Dic271⋊C4—22no—
90Dic271⋊C4—22no—
91Dic271⋊C4—22no—
92C22⋊Dic271—22no—
93C22⋊Dic271—22no—
94(C2×D271)⋊C4—22no—
95(C2×D271)⋊C4—22no—
96(C2×D271)⋊C4—22no—
97(C2×D271)⋊C4—22no—
98(C2×D271)⋊C4—22no—
99(C2×D271)⋊C4—22no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—22no—
102C4×Dic271—22no—
103C4×Dic271—22no—
104C24×C271C2311yescharacteristic
105C22×C4×C271—22no—
106C22×C4×C271—22no—
107C22×C4×C271—22no—
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114C4⋊C4×C271—22no—
115C4⋊C4×C271—22no—
116C22⋊C4×C271—22no—
117C22⋊C4×C271—22no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—21no—
129C22×D271—21no—
130C22×D271C2×D412yes—
131C22×D271C2×D412yes—
132Dic542—28no—
133Dic542—44no—
134D1084—28no—
135C22⋊D271—28no—
136C22⋊D271—28no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271C2×D412yes—
151C2×Dic271C2×D412yes—
152C2×Dic271C2×D412yes—
153C2×Dic271C2×D412yes—
154C2×Dic271#4112yescomplemented
155C4×D271—28no—
156C4×D271—28no—
157C23×C271—21no—
158C23×C271—21no—
159C23×C271—21no—
160C23×C271—21no—
161C23×C271—21no—
162C23×C271—21no—
163C23×C271C2×D412yes—
164C23×C271C2411yescharacteristic
165C2×C4×C271—28no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—21no—
177C2×C4×C271—21no—
178C2×C4×C271—21no—
179C2×C4×C271—21no—
180C2×C4×C271—21no—
181C2×C4×C271C2×D412yes—
182C2×C4×C271C2×D412yes—
183C2×C4×C271#4112yescomplemented
184C22×C271—22no—
185C22×C271—22no—
186C22×C271—22no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271—41no—
191C22×C271C22×D411yescharacteristic
192C22×C271C22×D411yescharacteristic
193C22×C271C22×D411yescharacteristic
194C22×C271C2×#4111yescharacteristic
195C22×C271C2×#4111yescharacteristic
196C22×C2712−1+411yescharacteristic
197C22×C2712−1+411yescharacteristic
198C2×D271—22no—
199C2×D271—22no—
200C2×D271—22no—
201C2×D271—22no—
202C2×D271—22no—
203C2×D271—22no—
204C2×D271—41no—
205C2×D271—41no—
206C2×D271—41no—
207C2×D271—41no—
208C2×D271—41no—
209C2×D271—41no—
210C2×D271—41no—
211C2×D271—41no—
212C4×C271—22no—
213C4×C271—22no—
214C4×C271—22no—
215C4×C271—41no—
216C4×C271—41no—
217C4×C271—41no—
218Dic271—22no—
219Dic271—22no—
220Dic271—41no—
221Dic271—41no—
222Dic271—41no—
223Dic271#12014yes—
224C2×C271—42no—
225C2×C271C2×#12011yescharacteristic
226C2×C271#53211yescharacteristic
227C2×C271#53611yescharacteristic
228C2×C271#54711yescharacteristic
229C2×C271#54711yescharacteristic
230C2×C271#54811yescharacteristic
231C2×C271#54811yescharacteristic
232D271—42no—
233D271—42no—
234C271#243211yescharacteristic, complemented, Sylow
235#2432—2711nomaximal, Sylow
236C22×#41—2711no—
237C2×C4⋊Q8—2711no—
238C2×#124—2711no—
239C2×C22⋊Q8—2711no—
240#394—2711no—
241#394—2711no—
242#394—2711no—
243#394—2711no—
244#391—2711no—
245#390—2711no—
246#385—2711no—
247#385—2711no—
248#385—2711no—
249#385D27111yescharacteristic, complemented
250C4×C22⋊C4—2711no—
251C2×#41—5424no—
252C2×#41—5424no—
253C2×#41—5424no—
254C22×Q8—2711no—
255C22×Q8—2711no—
256C22×D4—2711no—
257C22×D4—2711no—
258C22×D4—2711no—
259C23×C4—2711no—
260C23×C4—2711no—
261C23×C4C2×D27111yescharacteristic
262C4⋊Q8—5424no—
263#124—5424no—
264C22⋊Q8—5424no—
265C2×C4⋊C4—2711no—
266C2×C4⋊C4—2711no—
267C2×C4⋊C4—2711no—
268C2×C4⋊C4—2711no—
269C2×C4⋊C4—2711no—
270C2×C4⋊C4C2×D27111yescharacteristic
271C2×C4⋊C4C2×D27111yescharacteristic
272C2×C22⋊C4—2711no—
273C2×C22⋊C4—2711no—
274C2×C22⋊C4—2711no—
275C2×C22⋊C4—2711no—
276C2×C22⋊C4—2711no—
277C2×C22⋊C4—2711no—
278C2×C22⋊C4—2711no—
279C2×C22⋊C4—2711no—
280C2×C22⋊C4—2711no—
281C2×C22⋊C4—2711no—
282C2×C22⋊C4C2×D27111yescharacteristic
283C2×C22⋊C4C2×D27111yescharacteristic
284C2×C42—2711no—
285C2×C42—2711no—
286#95—2711no—
287#95—2711no—
288#95—2711no—
289#95—2711no—
290#95C2×D27111yescharacteristic
291#95C2×D27111yescharacteristic
292C24—2711no—
293C24—2711no—
294C24C22×D27111yescharacteristic
295#41—54232no—
296C2×Q8—2714no—
297C2×Q8—5422no—
298C2×Q8—5422no—
299C2×Q8—5422no—
300C2×Q8—5422no—
301C2×Q8—5422no—
302C2×D4—2714no—
303C2×D4—5422no—
304C2×D4—5422no—
305C2×D4—5422no—
306C2×D4—5422no—
307C2×D4—5422no—
308C2×D4—5422no—
309C2×D4—5422no—
310C2×D4—5422no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—2711no—
315C22×C4—2711no—
316C22×C4—2711no—
317C22×C4—2714no—
318C22×C4—2711no—
319C22×C4—2711no—
320C22×C4—2711no—
321C22×C4—5422no—
322C22×C4—5422no—
323C22×C4—5422no—
324C22×C4—5422no—
325C22×C4—5422no—
326C22×C4C22×D27111yescharacteristic
327C22×C4C22×D27111yescharacteristic
328C22×C4C22×D27111yescharacteristic
329C22×C4C22×D27111yescharacteristic
330C22×C4C22×D27111yescharacteristic
331C22×C4C22×D27111yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—5422no—
335C4⋊C4—5422no—
336C4⋊C4—5422no—
337C4⋊C4—5422no—
338C4⋊C4—5422no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—5422no—
342C22⋊C4—5422no—
343C22⋊C4—5422no—
344C22⋊C4—5422no—
345C22⋊C4—5422no—
346C22⋊C4—5422no—
347C22⋊C4—5422no—
348C22⋊C4—5422no—
349C22⋊C4—5422no—
350C22⋊C4—5422no—
351C42—5422no—
352C42—5422no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—2712no—
360C23—2712no—
361C23—5421no—
362C23—5421no—
363C23—5421no—
364C23—5421no—
365C23—5421no—
366C23—5421no—
367C23—5421no—
368C23—5421no—
369C23—5421no—
370C23—5421no—
371C23—5421no—
372C23—5421no—
373C23D4×D27112yes—
374C23C23×D27111yescharacteristic
375Q8—5428no—
376Q8—10844no—
377D4—5428no—
378D4—5428no—
379D4—5428no—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—2712no—
397C2×C4—2712no—
398C2×C4—2712no—
399C2×C4—2712no—
400C2×C4—2712no—
401C2×C4—5428no—
402C2×C4—5428no—
403C2×C4—5421no—
404C2×C4—5421no—
405C2×C4—5421no—
406C2×C4—5421no—
407C2×C4—5421no—
408C2×C4—5421no—
409C2×C4—5421no—
410C2×C4—5421no—
411C2×C4—5421no—
412C2×C4—5421no—
413C2×C4—5421no—
414C2×C4—5421no—
415C2×C4—5421no—
416C2×C4D4×D27112yes—
417C2×C4D4×D27112yes—
418C2×C4#1686465412yescomplemented
419C22—22no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—5422no—
427C22—5422no—
428C22—5422no—
429C22—5422no—
430C22—5422no—
431C22—5422no—
432C22—10841no—
433C22—10841no—
434C22—10841no—
435C22—10841no—
436C22—10841no—
437C22—10841no—
438C22—10841no—
439C22—10841no—
440C22C2×D4×D27111yescharacteristic
441C22C2×D4×D27111yescharacteristic
442C22C2×D4×D27111yescharacteristic
443C22#41×D27111yescharacteristic
444C22C2×#1686465411yescharacteristic
445C22#2257620711yescharacteristic
446C22#2257620811yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—2714no—
454C4—5422no—
455C4—5422no—
456C4—10841no—
457C4—10841no—
458C4—10841no—
459C2—42no—
460C2—10842no—
461C2—10842no—
462C2#120×D27111yescharacteristic
463C2#3177444411yescharacteristic
464C2#3177447011yescharacteristic
465C2#3177454511yescharacteristic
466C2#3177454811yescharacteristic
467C2#3177455611yescharacteristic
468C2#3177455711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#385×C271C2
prime radicalC22×#16864654C2
hypercenter#385D271
hypocenterC271#2432
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#22576159C2semidirect
3. C2×#22576146C2semidirect
4. C2×#22576138C2semidirect
5. #31773815C2semidirect
6. #31773814C2semidirect
7. #31773813C2semidirect
8. #31773812C2semidirect
9. #31773801C2semidirect
10. #31773797C2semidirect
11. #31773771C2semidirect
12. #31773770C2semidirect
13. #31773768C2semidirect
14. #31773744C2semidirect
15. #385×C271C2semidirect
154. C2×Dic271#41semidirect
183. C2×C4×C271#41semidirect
234. C271#2432semidirect
249. #385D271semidirect
418. C2×C4#16864654semidirect

Maximal subgroups

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

subgroupindex
C22×#168646542
C2×#225761592
C2×#225761462
C2×#225761382
#317738152
#317738142
#317738132
#317738122
#317738012
#317737972
#317737712
#317737702
#317737682
#317737442
#385×C2712
#2432271