Finite Group Databasebeta

← #46840108

Subgroups — #46840108

Order 34688.

610 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576132C211yesmaximal, characteristic
2#31773835C211yesmaximal, characteristic, complemented
3#31773825C211yesmaximal, characteristic, complemented
4#31773823C211yesmaximal, characteristic, complemented
5#31773812C211yesmaximal, characteristic, complemented
6#31773788C211yesmaximal, characteristic, complemented
7#31773781C211yesmaximal, characteristic, complemented
8#31773777C211yesmaximal, characteristic, complemented
9#31773776C211yesmaximal, characteristic, complemented
10#31773776C211yesmaximal, characteristic, complemented
11#31773774C211yesmaximal, characteristic, complemented
12#31773769C211yesmaximal, characteristic, complemented
13#31773760C211yesmaximal, characteristic, complemented
14C2×C271⋊#95C211yesmaximal, characteristic, complemented
15C2×C4×D4×C271C211yesmaximal, characteristic, complemented
16C22×C22⋊D271C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19C23×Dic271C2211yescharacteristic
20#22576132—24no—
21C2×Dic271⋊C4C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×C22⋊Dic271C2211yescharacteristic
25C2×C22⋊Dic271C2211yescharacteristic
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×(C2×D271)⋊C4C2211yescharacteristic
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C271⋊#95C2211yescharacteristic
34C271⋊#95C2211yescharacteristic
35C271⋊#95C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C22×D4×C271C2211yescharacteristic
46C23×C4×C271C2211yescharacteristic
47C23×C4×C271C2211yescharacteristic
48C4×D4×C271—24no—
49C2×C4⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C42×C271C2211yescharacteristic
53C23×D271C2311yescharacteristic
54C2×C22⋊D271—22no—
55C2×C22⋊D271—22no—
56C2×C22⋊D271—22no—
57C2×C22⋊D271—22no—
58C2×C22⋊D271—22no—
59C2×C22⋊D271—41no—
60C22×Dic271—22no—
61C22×Dic271—22no—
62C22×Dic271—22no—
63C22×Dic271—41no—
64C22×Dic271—41no—
65C22×Dic271—41no—
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73Dic271⋊C4—22no—
74Dic271⋊C4—22no—
75Dic271⋊C4—41no—
76C22⋊Dic271—22no—
77C22⋊Dic271—22no—
78C22⋊Dic271—41no—
79C22⋊Dic271—41no—
80(C2×D271)⋊C4—22no—
81(C2×D271)⋊C4—22no—
82(C2×D271)⋊C4—41no—
83(C2×D271)⋊C4—41no—
84(C2×D271)⋊C4—41no—
85C24×C271C2311yescharacteristic
86C24×C271C2311yescharacteristic
87C2×D4×C271—22no—
88C2×D4×C271—22no—
89C2×D4×C271D414yes—
90C22×C4×C271—22no—
91C22×C4×C271—22no—
92C22×C4×C271—22no—
93C22×C4×C271—22no—
94C22×C4×C271—41no—
95C22×C4×C271—41no—
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C4⋊C4×C271—22no—
102C22⋊C4×C271—22no—
103C22⋊C4×C271—22no—
104C42×C271—22no—
105C22×D271—21no—
106C22×D271—21no—
107C22×D271—21no—
108C22×D271—21no—
109C22×D271—21no—
110C22×D271—21no—
111C22×D271—21no—
112C22⋊D271—44no—
113C22⋊D271—44no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—44no—
135C2×Dic271—44no—
136C2×Dic271#4112yes—
137C23×C271—21no—
138C23×C271—21no—
139C23×C271—21no—
140C23×C271—21no—
141C23×C271—21no—
142C23×C271—21no—
143C23×C271—21no—
144C23×C271—21no—
145C23×C271—21no—
146C23×C271—21no—
147C23×C271C2×D412yes—
148C23×C271C2×D412yes—
149C23×C271#4112yes—
150C23×C271#4112yes—
151C23×C271C2411yescharacteristic
152D4×C271—28no—
153C2×C4×C271—21no—
154C2×C4×C271—21no—
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—44no—
167C2×C4×C271—44no—
168C2×C4×C271C2×D412yes—
169C2×C4×C271#4112yes—
170C22×C271—22no—
171C22×C271—22no—
172C22×C271—22no—
173C22×C271—22no—
174C22×C271—22no—
175C22×C271—22no—
176C22×C271—22no—
177C22×C271—22no—
178C22×C271—22no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271#12314yes—
184C22×C271C22×D411yescharacteristic
185C22×C271C2×#4111yescharacteristic
186C22×C271C2×#4111yescharacteristic
187C22×C271C2×#4111yescharacteristic
188C22×C271C2×#4111yescharacteristic
189C22×C271H(5,𝔽2)11yescharacteristic
190C22×C271H(5,𝔽2)11yescharacteristic
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C4×C271—22no—
199C4×C271—22no—
200C4×C271—41no—
201C4×C271—41no—
202C4×C271—41no—
203Dic271—22no—
204Dic271—41no—
205Dic271—41no—
206Dic271—41no—
207Dic271—41no—
208Dic271—41no—
209Dic271—41no—
210C2×C271—24no—
211C2×C271—42no—
212C2×C271C2×#12311yescharacteristic
213C2×C271#52911yescharacteristic
214C2×C271#53811yescharacteristic
215C2×C271#54611yescharacteristic
216C2×C271#54611yescharacteristic
217C2×C271#55311yescharacteristic
218C2×C271#55311yescharacteristic
219D271—81no—
220C271#244411yescharacteristic, complemented, Sylow
221#2444—2711nomaximal, Sylow
222C2×#121—2711no—
223C2×C4×D4D27111yescharacteristic, complemented
224#400—2711no—
225#397—2711no—
226#397—2711no—
227#394—2711no—
228#388—2711no—
229#387—2711no—
230#386—2711no—
231#386—2711no—
232#386—2711no—
233#386—2711no—
234#385—2711no—
235#381—2711no—
236C2×#95—2711no—
237C22×D4—2711no—
238C22×D4—2711no—
239C22×D4C2×D27111yescharacteristic
240C23×C4—2711no—
241C23×C4—2711no—
242C23×C4C2×D27111yescharacteristic
243C23×C4C2×D27111yescharacteristic
244#121—5424no—
245C4×D4—24no—
246C2×C4⋊C4—2711no—
247C2×C4⋊C4—2711no—
248C2×C4⋊C4—2711no—
249C2×C4⋊C4C2×D27111yescharacteristic
250C2×C22⋊C4—2711no—
251C2×C22⋊C4—2711no—
252C2×C22⋊C4—2711no—
253C2×C22⋊C4—2711no—
254C2×C22⋊C4—2711no—
255C2×C22⋊C4—2711no—
256C2×C22⋊C4—2711no—
257C2×C22⋊C4—2711no—
258C2×C22⋊C4—2711no—
259C2×C22⋊C4C2×D27111yescharacteristic
260C2×C22⋊C4C2×D27111yescharacteristic
261C2×C42C2×D27111yescharacteristic
262#95—2711no—
263#95—2711no—
264#95—2711no—
265#95—2711no—
266#95—2711no—
267#95—2711no—
268#95—2711no—
269#95—2711no—
270#95—2711no—
271#95—2711no—
272#95—2711no—
273#95—2711no—
274C24—2711no—
275C24C22×D27111yescharacteristic
276C24C22×D27111yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—5422no—
280C2×D4—5422no—
281C2×D4—5422no—
282C2×D4—5422no—
283C2×D4—5422no—
284C2×D4—10841no—
285C2×D4C22⋊D27114yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—2711no—
293C22×C4—2711no—
294C22×C4—2711no—
295C22×C4—2711no—
296C22×C4—2711no—
297C22×C4—2711no—
298C22×C4—2711no—
299C22×C4—5422no—
300C22×C4—5422no—
301C22×C4—5422no—
302C22×C4—10841no—
303C22×C4—10841no—
304C22×C4—10841no—
305C22×C4C22×D27111yescharacteristic
306C22×C4C22×D27111yescharacteristic
307C22×C4C22×D27111yescharacteristic
308C22×C4C22×D27111yescharacteristic
309C22×C4C22×D27111yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—5422no—
312C4⋊C4—5422no—
313C4⋊C4—10841no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—5422no—
317C22⋊C4—5422no—
318C22⋊C4—5422no—
319C22⋊C4—5422no—
320C22⋊C4—10841no—
321C22⋊C4—10841no—
322C22⋊C4—10841no—
323C22⋊C4—10841no—
324C22⋊C4—10841no—
325C42—22no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—5421no—
337C23—5421no—
338C23—5421no—
339C23—5421no—
340C23—5421no—
341C23—5421no—
342C23—5421no—
343C23C2×C22⋊D27112yes—
344C23C2×C22⋊D27112yes—
345C23#1686465412yes—
346C23#1686465412yes—
347C23C23×D27111yescharacteristic
348D4—28no—
349D4—10844no—
350D4—10844no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—2712no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—5421no—
373C2×C4—5421no—
374C2×C4—5421no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—5421no—
380C2×C4—5421no—
381C2×C4—5421no—
382C2×C4—5421no—
383C2×C4—5421no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—5421no—
387C2×C4—10844no—
388C2×C4—10844no—
389C2×C4C2×C22⋊D27112yes—
390C2×C4#1686465312yes—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—10841no—
410C22—10841no—
411C22#2257614314yes—
412C22C22×C22⋊D27111yescharacteristic
413C22#41×D27111yescharacteristic
414C22C2×#1686465311yescharacteristic
415C22C2×#1686465411yescharacteristic
416C22C2×#1686465411yescharacteristic
417C22#2257620511yescharacteristic
418C22#2257620611yescharacteristic
419C4—22no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—5422no—
425C4—10841no—
426C4—10841no—
427C4—10841no—
428C4—10841no—
429C4—10841no—
430C4—10841no—
431C2—24no—
432C2—42no—
433C2—21681no—
434C2C2×#2257614311yescharacteristic
435C2#3177442711yescharacteristic
436C2#3177448011yescharacteristic
437C2#3177453511yescharacteristic
438C2#3177454111yescharacteristic
439C2#3177459311yescharacteristic
440C2#3177459611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×C4×D4×C271C2
prime radicalC2×#22576132C2
hypercenterC2×C4×D4D271
hypocenterC271#2444
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #31773835C2semidirect
3. #31773825C2semidirect
4. #31773823C2semidirect
5. #31773812C2semidirect
6. #31773788C2semidirect
7. #31773781C2semidirect
8. #31773777C2semidirect
9. #31773776C2semidirect
10. #31773776C2semidirect
11. #31773774C2semidirect
12. #31773769C2semidirect
13. #31773760C2semidirect
14. C2×C271⋊#95C2semidirect
15. C2×C4×D4×C271C2semidirect
220. C271#2444semidirect
223. C2×C4×D4D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761322
#317738352
#317738252
#317738232
#317738122
#317737882
#317737812
#317737772
#317737762
#317737742
#317737692
#317737602
C2×C271⋊#952
C2×C4×D4×C2712
#2444271