Finite Group Databasebeta

← #46839052

Subgroups — #46839052

Order 34688.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576133C211yesmaximal, characteristic
2C2×(C22⋊D271)⋊C4C211yesmaximal, characteristic, complemented
3C2×#22576118C211yesmaximal, characteristic, complemented
4#31773799C211yesmaximal, characteristic, complemented
5#31773797C211yesmaximal, characteristic, complemented
6#31773797C211yesmaximal, characteristic, complemented
7#31773794C211yesmaximal, characteristic, complemented
8#31773787C211yesmaximal, characteristic, complemented
9#31773775C211yesmaximal, characteristic, complemented
10#31773774C211yesmaximal, characteristic, complemented
11#31773772C211yesmaximal, characteristic, complemented
12#31773771C211yesmaximal, characteristic, complemented
13#31773756C211yesmaximal, characteristic, complemented
14C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
15#388×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C23×Dic271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22#22576133C418yescomplemented
23(C22⋊D271)⋊C4—24no—
24#22576118—24no—
25C2×C4⋊Dic271C2211yescharacteristic
26C2×C4⋊Dic271C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×C4×Dic271C2211yescharacteristic
41C2×C4×Dic271C2211yescharacteristic
42C2×C4×Dic271C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C2×C4⋊C4×C271C2211yescharacteristic
48C2×C22⋊C4×C271C2211yescharacteristic
49C2×C22⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C42×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53#95×C271C2211yescharacteristic
54C23×D271C2311yescharacteristic
55C23×D271C2311yescharacteristic
56C2×D1084—41no—
57C2×D1084—41no—
58C2×D1084C2×C414yescomplemented
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—41no—
62C2×C22⋊D271—41no—
63C2×C22⋊D271C2×C414yes—
64C2×C22⋊D271C2×C414yescomplemented
65C22×Dic271—22no—
66C22×Dic271—41no—
67C22×Dic271—41no—
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C22×Dic271C2311yescharacteristic
74C2×C4×D271—22no—
75C2×C4×D271—22no—
76C2×C4×D271—22no—
77C2×C4×D271—41no—
78C2×C4×D271—41no—
79C2×C4×D271C2×C414yescomplemented
80C4⋊Dic271—22no—
81C4⋊Dic271—41no—
82Dic271⋊C4—22no—
83Dic271⋊C4—22no—
84Dic271⋊C4—41no—
85Dic271⋊C4C2×C414yescomplemented
86C22⋊Dic271—41no—
87(C2×D271)⋊C4—22no—
88(C2×D271)⋊C4—22no—
89(C2×D271)⋊C4—22no—
90(C2×D271)⋊C4—41no—
91(C2×D271)⋊C4—41no—
92(C2×D271)⋊C4—41no—
93(C2×D271)⋊C4—41no—
94(C2×D271)⋊C4C2×C414yescomplemented
95C4×Dic271—22no—
96C4×Dic271—41no—
97C4×Dic271D414yescomplemented
98C24×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103C22×C4×C271C2311yescharacteristic
104C22×C4×C271C2311yescharacteristic
105C4⋊C4×C271—22no—
106C22⋊C4×C271—22no—
107C22⋊C4×C271—41no—
108C22⋊C4×C271C2×C414yescomplemented
109C42×C271—41no—
110C22×D271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—21no—
116C22×D271—21no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271C22×C412yes—
123C22×D271C22×C412yes—
124D1084—44no—
125C22⋊D271—28no—
126C22⋊D271—44no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—44no—
141C2×Dic271C22×C412yes—
142C2×Dic271C22×C412yes—
143C2×Dic271C2×D412yes—
144C2×Dic271C2×D412yes—
145C2×Dic271#4112yes—
146C4×D271—28no—
147C4×D271—44no—
148C23×C271—21no—
149C23×C271—21no—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—21no—
153C23×C271—21no—
154C23×C271C22×C412yes—
155C23×C271C2411yescharacteristic
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271C22×C412yes—
172C2×C4×C271C22×C412yes—
173C2×C4×C271C2×D412yes—
174C22×C271—22no—
175C22×C271—22no—
176C22×C271—22no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271C23×C411yescharacteristic
182C22×C271C22×D411yescharacteristic
183C22×C271C2×#4111yescharacteristic
184C22×C271H(5,𝔽2)11yescharacteristic
185C22×C271H(5,𝔽2)11yescharacteristic
186C22×C271H(5,𝔽2)11yescharacteristic
187C22×C2712−1+411yescharacteristic
188C2×D271—22no—
189C2×D271—22no—
190C2×D271—22no—
191C2×D271—22no—
192C2×D271—22no—
193C2×D271—22no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C4×C271—22no—
203C4×C271—22no—
204C4×C271—22no—
205C4×C271—41no—
206C4×C271—41no—
207C4×C271—41no—
208Dic271—22no—
209Dic271—22no—
210Dic271—41no—
211Dic271—41no—
212Dic271—41no—
213Dic271C4×D414yes—
214C2×C271—42no—
215C2×C271C2×C4×D411yescharacteristic
216C2×C271#51811yescharacteristic
217C2×C271#52011yescharacteristic
218C2×C271#53511yescharacteristic
219C2×C271#53711yescharacteristic
220C2×C271#54011yescharacteristic
221C2×C271#54211yescharacteristic
222D271—42no—
223D271—42no—
224C271#232411yescharacteristic, complemented, Sylow
225#2324—2711nomaximal, Sylow
226C2×#121—2711no—
227C2×C4×D4—2711no—
228C2×#117—2711no—
229#390—2711no—
230#390—2711no—
231#390—2711no—
232#389—2711no—
233#388—2711no—
234#388D27111yescharacteristic, complemented
235#386—2711no—
236#386—2711no—
237#385—2711no—
238#385—2711no—
239#380—2711no—
240C4×C4⋊C4—2711no—
241C22×D4—2711no—
242C22×D4—2711no—
243C22×D4—2711no—
244C23×C4—2711no—
245C23×C4—2711no—
246C23×C4—2711no—
247#121—2718no—
248C4×D4—5424no—
249#117—5424no—
250C2×C4⋊C4—2711no—
251C2×C4⋊C4—2711no—
252C2×C4⋊C4—2711no—
253C2×C4⋊C4—2711no—
254C2×C4⋊C4—2711no—
255C2×C4⋊C4—2711no—
256C2×C4⋊C4C2×D27111yescharacteristic
257C2×C22⋊C4—2711no—
258C2×C22⋊C4—2711no—
259C2×C22⋊C4—2711no—
260C2×C22⋊C4—2711no—
261C2×C22⋊C4—2711no—
262C2×C22⋊C4—2711no—
263C2×C22⋊C4—2711no—
264C2×C22⋊C4—2711no—
265C2×C22⋊C4—2711no—
266C2×C22⋊C4C2×D27111yescharacteristic
267C2×C22⋊C4C2×D27111yescharacteristic
268C2×C22⋊C4C2×D27111yescharacteristic
269C2×C42—2711no—
270C2×C42—2711no—
271C2×C42—2711no—
272C2×C42C2×D27111yescharacteristic
273#95—2711no—
274#95—2711no—
275#95—2711no—
276#95—2711no—
277#95C2×D27111yescharacteristic
278#95C2×D27111yescharacteristic
279C24—2711no—
280C24—2711no—
281C24C22×D27111yescharacteristic
282C2×D4—2714no—
283C2×D4—2714no—
284C2×D4—2714no—
285C2×D4—5422no—
286C2×D4—5422no—
287C2×D4—10841no—
288C2×D4—10841no—
289C2×D4—10841no—
290C2×D4—10841no—
291C22×C4—2711no—
292C22×C4—2711no—
293C22×C4—2711no—
294C22×C4—2714no—
295C22×C4—2711no—
296C22×C4—2711no—
297C22×C4—2711no—
298C22×C4—5422no—
299C22×C4—5422no—
300C22×C4—5422no—
301C22×C4—5422no—
302C22×C4—10841no—
303C22×C4—10841no—
304C22×C4—10841no—
305C22×C4—10841no—
306C22×C4C22×D27111yescharacteristic
307C22×C4C22×D27111yescharacteristic
308C22×C4C22×D27111yescharacteristic
309C22×C4C22×D27111yescharacteristic
310C22×C4C22×D27111yescharacteristic
311C22×C4C22×D27111yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—2714no—
314C4⋊C4—5422no—
315C4⋊C4—5422no—
316C4⋊C4—5422no—
317C4⋊C4—10841no—
318C4⋊C4—10841no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—2714no—
322C22⋊C4—5422no—
323C22⋊C4—5422no—
324C22⋊C4—5422no—
325C22⋊C4—10841no—
326C22⋊C4—10841no—
327C22⋊C4—10841no—
328C22⋊C4—10841no—
329C22⋊C4—10841no—
330C22⋊C4C4×D27114yescomplemented
331C42—41no—
332C42—2714no—
333C42—5422no—
334C42—10841no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2712no—
342C23—2712no—
343C23—5421no—
344C23—5421no—
345C23—5421no—
346C23—5421no—
347C23—5421no—
348C23—5421no—
349C23—5421no—
350C23—5421no—
351C23—5421no—
352C23—5421no—
353C23—5421no—
354C23—5421no—
355C23C2×C4×D27112yes—
356C23C23×D27111yescharacteristic
357D4—5428no—
358D4—10844no—
359D4—10844no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—2712no—
376C2×C4—2712no—
377C2×C4—2712no—
378C2×C4—2712no—
379C2×C4—2712no—
380C2×C4—5428no—
381C2×C4—5421no—
382C2×C4—5421no—
383C2×C4—5421no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—5421no—
387C2×C4—5421no—
388C2×C4—5421no—
389C2×C4—5421no—
390C2×C4—5421no—
391C2×C4—5421no—
392C2×C4—5421no—
393C2×C4—5421no—
394C2×C4—10844no—
395C2×C4—10844no—
396C2×C4C2×C4×D27112yes—
397C2×C4C2×C4×D27112yes—
398C2×C4D4×D27112yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—5422no—
407C22—5422no—
408C22—5422no—
409C22—5422no—
410C22—5422no—
411C22—5422no—
412C22—10841no—
413C22—10841no—
414C22—10841no—
415C22—10841no—
416C22—10841no—
417C22—10841no—
418C22—10841no—
419C22—10841no—
420C22C22×C4×D27111yescharacteristic
421C22C2×D4×D27111yescharacteristic
422C22#41×D27111yescharacteristic
423C22#2257620511yescharacteristic
424C22#2257620611yescharacteristic
425C22#2257620611yescharacteristic
426C22#2257620711yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—2714no—
434C4—5422no—
435C4—5422no—
436C4—10841no—
437C4—10841no—
438C4—10841no—
439C2—42no—
440C2—10842no—
441C2—10842no—
442C2C4×D4×D27111yescharacteristic
443C2#3177436911yescharacteristic
444C2#3177437711yescharacteristic
445C2#3177446511yescharacteristic
446C2#3177447411yescharacteristic
447C2#3177449911yescharacteristic
448C2#3177451311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#388×C271C2
socle, cosocleC23×C271C24
prime radicalC2×#22576133C2
hypercenter#388D271
hypocenterC271#2324
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D271)⋊C4C2semidirect
3. C2×#22576118C2semidirect
4. #31773799C2semidirect
5. #31773797C2semidirect
6. #31773797C2semidirect
7. #31773794C2semidirect
8. #31773787C2semidirect
9. #31773775C2semidirect
10. #31773774C2semidirect
11. #31773772C2semidirect
12. #31773771C2semidirect
13. #31773756C2semidirect
14. C4×Dic271⋊C4C2semidirect
15. #388×C271C2semidirect
22. #22576133C4semidirect
58. C2×D1084C2×C4semidirect
64. C2×C22⋊D271C2×C4semidirect
79. C2×C4×D271C2×C4semidirect
85. Dic271⋊C4C2×C4semidirect
94. (C2×D271)⋊C4C2×C4semidirect
97. C4×Dic271D4semidirect
108. C22⋊C4×C271C2×C4semidirect
224. C271#2324semidirect
234. #388D271semidirect
330. C22⋊C4C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761332
C2×(C22⋊D271)⋊C42
C2×#225761182
#317737992
#317737972
#317737942
#317737872
#317737752
#317737742
#317737722
#317737712
#317737562
C4×Dic271⋊C42
#388×C2712
#2324271