Finite Group Databasebeta

← #46839915

Subgroups — #46839915

Order 34688.

666 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576143C211yesmaximal, characteristic, complemented
2C2×#22576143C211yesmaximal, characteristic, complemented
3C2×#22576134C211yesmaximal, characteristic
4C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
5#31773831C211yesmaximal, characteristic, complemented
6#31773826C211yesmaximal, characteristic, complemented
7#31773821C211yesmaximal, characteristic, complemented
8#31773815C211yesmaximal, characteristic, complemented
9#31773794C211yesmaximal, characteristic, complemented
10#31773786C211yesmaximal, characteristic, complemented
11#31773776C211yesmaximal, characteristic, complemented
12#31773775C211yesmaximal, characteristic, complemented
13#31773770C211yesmaximal, characteristic, complemented
14#31773756C211yesmaximal, characteristic, complemented
15C2×#95×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20#22576143—24no—
21#22576143—24no—
22#22576134—24no—
23C4×C22⋊D271—24no—
24C2×C4⋊Dic271C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×Dic271⋊C4C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
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×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic
43C2×C4×Dic271C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C23×C4×C271C2211yescharacteristic
49C23×C4×C271C2211yescharacteristic
50C23×C4×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53#95×C271C2211yescharacteristic
54#95×C271C2211yescharacteristic
55C23×D271C2311yescharacteristic
56C23×D271C2311yescharacteristic
57C2×D1084—22no—
58C2×D1084—22no—
59C2×D1084—41no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—22no—
62C2×C22⋊D271—22no—
63C2×C22⋊D271—22no—
64C2×C22⋊D271—22no—
65C2×C22⋊D271D414yes—
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C2×C4×D271—22no—
73C2×C4×D271—22no—
74C2×C4×D271—41no—
75C4⋊Dic271—22no—
76Dic271⋊C4—22no—
77Dic271⋊C4—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—22no—
80Dic271⋊C4—22no—
81Dic271⋊C4—22no—
82Dic271⋊C4—41no—
83C22⋊Dic271—22no—
84C22⋊Dic271D414yescomplemented
85(C2×D271)⋊C4—22no—
86(C2×D271)⋊C4—22no—
87(C2×D271)⋊C4—22no—
88(C2×D271)⋊C4—22no—
89(C2×D271)⋊C4—22no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—41no—
94C4×Dic271—22no—
95C24×C271C2311yescharacteristic
96C22×C4×C271—22no—
97C22×C4×C271—22no—
98C22×C4×C271—22no—
99C22×C4×C271—22no—
100C22×C4×C271—22no—
101C22×C4×C271—22no—
102C22×C4×C271—22no—
103C22×C4×C271—41no—
104C22×C4×C271—41no—
105C22×C4×C271C2311yescharacteristic
106C22×C4×C271C2311yescharacteristic
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
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×D271—21no—
123C22×D271—21no—
124C22×D271C2×D412yes—
125D1084—44no—
126C22⋊D271—28no—
127C22⋊D271—44no—
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—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271C2×D412yes—
144C2×Dic271C2×D412yes—
145C2×Dic271C2×D412yes—
146C4×D271—44no—
147C23×C271—21no—
148C23×C271—21no—
149C23×C271—21no—
150C23×C271—21no—
151C23×C271C2×D412yes—
152C23×C271C2×D412yes—
153C23×C271#4112yes—
154C23×C271C2411yescharacteristic
155C2×C4×C271—21no—
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×C271—44no—
172C2×C4×C271—44no—
173C2×C4×C271—44no—
174C2×C4×C271#4112yes—
175C2×C4×C271#4112yes—
176C22×C271—22no—
177C22×C271—22no—
178C22×C271—22no—
179C22×C271—22no—
180C22×C271—22no—
181C22×C271—22no—
182C22×C271#12114yes—
183C22×C271C22×D411yescharacteristic
184C22×C271C22×D411yescharacteristic
185C22×C271C2×#4111yescharacteristic
186C22×C271C2×#4111yescharacteristic
187C22×C271C2×#4111yescharacteristic
188C22×C271H(5,𝔽2)11yescharacteristic
189C22×C2712−1+411yescharacteristic
190C2×D271—22no—
191C2×D271—22no—
192C2×D271—22no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C2×D271—41no—
203C2×D271—41no—
204C4×C271—22no—
205C4×C271—22no—
206C4×C271—41no—
207C4×C271—41no—
208C4×C271—41no—
209C4×C271—41no—
210Dic271—22no—
211Dic271—22no—
212Dic271—22no—
213Dic271—41no—
214Dic271—41no—
215Dic271—41no—
216C2×C271—24no—
217C2×C271C2×#12111yescharacteristic
218C2×C271#52911yescharacteristic
219C2×C271#53911yescharacteristic
220C2×C271#54611yescharacteristic
221C2×C271#54611yescharacteristic
222C2×C271#54711yescharacteristic
223C2×C271#54711yescharacteristic
224D271—42no—
225D271—81no—
226C271#242511yescharacteristic, complemented, Sylow
227#2425—2711nomaximal, Sylow
228C2×#123—2711no—
229C2×#123—2711no—
230C2×#121—2711no—
231C2×C4×D4—2711no—
232#398—2711no—
233#397—2711no—
234#396—2711no—
235#394—2711no—
236#389—2711no—
237#388—2711no—
238#386—2711no—
239#386—2711no—
240#385—2711no—
241#380—2711no—
242C2×#95D27111yescharacteristic, complemented
243C22×D4—2711no—
244C22×D4—2711no—
245C22×D4—2711no—
246C23×C4—2711no—
247C23×C4C2×D27111yescharacteristic
248C23×C4C2×D27111yescharacteristic
249C23×C4C2×D27111yescharacteristic
250#123—5424no—
251#123—5424no—
252#121—5424no—
253C4×D4—5424no—
254C2×C4⋊C4—2711no—
255C2×C4⋊C4—2711no—
256C2×C4⋊C4—2711no—
257C2×C4⋊C4—2711no—
258C2×C4⋊C4—2711no—
259C2×C4⋊C4—2711no—
260C2×C4⋊C4—2711no—
261C2×C4⋊C4—2711no—
262C2×C22⋊C4—2711no—
263C2×C22⋊C4—2711no—
264C2×C22⋊C4—2711no—
265C2×C22⋊C4—2711no—
266C2×C22⋊C4—2711no—
267C2×C22⋊C4—2711no—
268C2×C22⋊C4—2711no—
269C2×C22⋊C4—2711no—
270C2×C22⋊C4—2711no—
271C2×C22⋊C4—2711no—
272C2×C22⋊C4—2711no—
273C2×C42—2711no—
274#95—2711no—
275#95—2711no—
276#95—2711no—
277#95—2711no—
278#95C2×D27111yescharacteristic
279#95C2×D27111yescharacteristic
280#95C2×D27111yescharacteristic
281#95C2×D27111yescharacteristic
282C24—2711no—
283C24—2711no—
284C24C22×D27111yescharacteristic
285C2×D4—2714no—
286C2×D4—5422no—
287C2×D4—5422no—
288C2×D4—5422no—
289C2×D4—5422no—
290C2×D4—5422no—
291C2×D4—5422no—
292C2×D4—5422no—
293C2×D4—10841no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—2711no—
304C22×C4—2711no—
305C22×C4—2711no—
306C22×C4—2711no—
307C22×C4—2711no—
308C22×C4—2711no—
309C22×C4—5422no—
310C22×C4—5422no—
311C22×C4—10841no—
312C22×C4C22×D27111yescharacteristic
313C22×C4C22×D27111yescharacteristic
314C22×C4C22×D27111yescharacteristic
315C22×C4C22×D27111yescharacteristic
316C22×C4C22×D27111yescharacteristic
317C22×C4C22×D27111yescharacteristic
318C4⋊C4—5422no—
319C4⋊C4—5422no—
320C4⋊C4—5422no—
321C4⋊C4—5422no—
322C4⋊C4—5422no—
323C4⋊C4—5422no—
324C4⋊C4—5422no—
325C4⋊C4—10841no—
326C22⋊C4—2714no—
327C22⋊C4—5422no—
328C22⋊C4—5422no—
329C22⋊C4—5422no—
330C22⋊C4—5422no—
331C22⋊C4—5422no—
332C22⋊C4—5422no—
333C22⋊C4—5422no—
334C22⋊C4—5422no—
335C22⋊C4—5422no—
336C22⋊C4—10841no—
337C42—5422no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
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—
355C23—5421no—
356C23D4×D27112yes—
357C23D4×D27112yes—
358C23#1686465412yes—
359C23C23×D27111yescharacteristic
360D4—5428no—
361D4—10844no—
362D4—10844no—
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—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—2712no—
383C2×C4—2712no—
384C2×C4—2712no—
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—5421no—
395C2×C4—5421no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—5421no—
399C2×C4—5421no—
400C2×C4—10844no—
401C2×C4#1686465412yes—
402C2×C4#1686465412yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—5422no—
410C22—5422no—
411C22—5422no—
412C22—10841no—
413C22—10841no—
414C22—10841no—
415C22—10841no—
416C22—10841no—
417C22—10841no—
418C22—10841no—
419C22—10841no—
420C22—10841no—
421C22—10841no—
422C22—10841no—
423C22#2257613314yes—
424C22C2×D4×D27111yescharacteristic
425C22C2×D4×D27111yescharacteristic
426C22C2×#1686465411yescharacteristic
427C22C2×#1686465411yescharacteristic
428C22C2×#1686465411yescharacteristic
429C22#2257620511yescharacteristic
430C22#2257620811yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—5422no—
438C4—5422no—
439C4—5422no—
440C4—10841no—
441C4—10841no—
442C4—10841no—
443C2—24no—
444C2—10842no—
445C2—21681no—
446C2C2×#2257613311yescharacteristic
447C2#3177443111yescharacteristic
448C2#3177449411yescharacteristic
449C2#3177453711yescharacteristic
450C2#3177453711yescharacteristic
451C2#3177454811yescharacteristic
452C2#3177454811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#95×C271C2
prime radicalC2×#22576134C2
hypercenterC2×#95D271
hypocenterC271#2425
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C2×#22576143C2semidirect
2. C2×#22576143C2semidirect
4. C2×C4×C22⋊D271C2semidirect
5. #31773831C2semidirect
6. #31773826C2semidirect
7. #31773821C2semidirect
8. #31773815C2semidirect
9. #31773794C2semidirect
10. #31773786C2semidirect
11. #31773776C2semidirect
12. #31773775C2semidirect
13. #31773770C2semidirect
14. #31773756C2semidirect
15. C2×#95×C271C2semidirect
84. C22⋊Dic271D4semidirect
226. C271#2425semidirect
242. C2×#95D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761432
C2×#225761342
C2×C4×C22⋊D2712
#317738312
#317738262
#317738212
#317738152
#317737942
#317737862
#317737762
#317737752
#317737702
#317737562
C2×#95×C2712
#2425271