Finite Group Databasebeta

← #46839851

Subgroups — #46839851

Order 34688.

596 conjugacy classes of subgroups.

The subgroup lattice has 440 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
3#31773826C211yesmaximal, characteristic, complemented
4#31773814C211yesmaximal, characteristic, complemented
5#31773812C211yesmaximal, characteristic, complemented
6#31773797C211yesmaximal, characteristic, complemented
7#31773792C211yesmaximal, characteristic, complemented
8#31773775C211yesmaximal, characteristic, complemented
9#31773774C211yesmaximal, characteristic, complemented
10#31773761C211yesmaximal, characteristic, complemented
11#31773759C211yesmaximal, characteristic, complemented
12#31773758C211yesmaximal, characteristic, complemented
13#31773756C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic, complemented
15#397×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C23×Dic271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22#22576133—24no—
23(C22⋊D271)⋊C4—24no—
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×C4×Dic271C2211yescharacteristic
38C2×C4×Dic271C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C2×C4⋊C4×C271C2211yescharacteristic
47C2×C22⋊C4×C271C2211yescharacteristic
48C2×C22⋊C4×C271C2211yescharacteristic
49C2×C22⋊C4×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53C23×D271C2311yescharacteristic
54C23×D271C2311yescharacteristic
55C2×D1084—22no—
56C2×D1084—22no—
57C2×D1084—41no—
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—22no—
62C2×C22⋊D271—41no—
63C2×C22⋊D271D414yes—
64C22×Dic271—22no—
65C22×Dic271—22no—
66C22×Dic271—41no—
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C2×C4×D271—22no—
74C2×C4×D271—22no—
75C2×C4×D271—22no—
76C2×C4×D271—22no—
77C2×C4×D271—22no—
78C2×C4×D271—41no—
79Dic271⋊C4—22no—
80Dic271⋊C4—22no—
81Dic271⋊C4—22no—
82Dic271⋊C4—41no—
83C22⋊Dic271—41no—
84(C2×D271)⋊C4—22no—
85(C2×D271)⋊C4—22no—
86(C2×D271)⋊C4—22no—
87(C2×D271)⋊C4—22no—
88(C2×D271)⋊C4—41no—
89(C2×D271)⋊C4—41no—
90(C2×D271)⋊C4—41no—
91(C2×D271)⋊C4—41no—
92C4×Dic271—22no—
93C4×Dic271—22no—
94C24×C271C2311yescharacteristic
95C22×C4×C271C2311yescharacteristic
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C4⋊C4×C271—41no—
102C22⋊C4×C271—22no—
103C22⋊C4×C271—22no—
104C22⋊C4×C271—41no—
105C22×D271—21no—
106C22×D271—21no—
107C22×D271—21no—
108C22×D271—21no—
109C22×D271—21no—
110C22×D271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—21no—
116C22×D271—21no—
117C22×D271C2×D412yes—
118C22×D271#4112yes—
119D1084—44no—
120C22⋊D271—28no—
121C22⋊D271—44no—
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—21no—
135C2×Dic271—21no—
136C2×Dic271—44no—
137C2×Dic271C2×D412yes—
138C2×Dic271#4112yes—
139C2×Dic271#4112yes—
140C2×Dic271#4112yes—
141C4×D271—44no—
142C4×D271—44no—
143C23×C271—21no—
144C23×C271—21no—
145C23×C271—21no—
146C23×C271—21no—
147C23×C271—21no—
148C23×C271—21no—
149C23×C271C2×D412yes—
150C23×C271C2411yescharacteristic
151C2×C4×C271—21no—
152C2×C4×C271—21no—
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—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C22×C271—22no—
170C22×C271—22no—
171C22×C271—22no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271C22×D411yescharacteristic
177C22×C271C2×#4111yescharacteristic
178C22×C271C2×#4111yescharacteristic
179C22×C271C2×#4111yescharacteristic
180C22×C271C2×#4111yescharacteristic
181C22×C271H(5,𝔽2)11yescharacteristic
182C22×C271H(5,𝔽2)11yescharacteristic
183C2×D271—22no—
184C2×D271—22no—
185C2×D271—22no—
186C2×D271—22no—
187C2×D271—22no—
188C2×D271—22no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—41no—
197C4×C271—41no—
198C4×C271—41no—
199C4×C271—41no—
200C4×C271—41no—
201C4×C271—41no—
202C4×C271—41no—
203Dic271—22no—
204Dic271—41no—
205Dic271—41no—
206Dic271—41no—
207Dic271—41no—
208Dic271#12314yes—
209C2×C271—42no—
210C2×C271C2×#12311yescharacteristic
211C2×C271#52911yescharacteristic
212C2×C271#54011yescharacteristic
213C2×C271#54611yescharacteristic
214C2×C271#54611yescharacteristic
215C2×C271#55311yescharacteristic
216C2×C271#55511yescharacteristic
217D271—42no—
218D271—42no—
219C271#242011yescharacteristic, complemented, Sylow
220#2420—2711nomaximal, Sylow
221C2×#121—2711no—
222C2×C4×D4—2711no—
223#397—2711no—
224#397D27111yescharacteristic, complemented
225#394—2711no—
226#394—2711no—
227#390—2711no—
228#389—2711no—
229#386—2711no—
230#386—2711no—
231#382—2711no—
232#381—2711no—
233#381—2711no—
234#380—2711no—
235C4×C22⋊C4—2711no—
236C22×D4—2711no—
237C22×D4—2711no—
238C22×D4—2711no—
239C23×C4—2711no—
240C23×C4—2711no—
241C23×C4—2711no—
242#121—5424no—
243C4×D4—5424no—
244C2×C4⋊C4—2711no—
245C2×C4⋊C4—2711no—
246C2×C4⋊C4—2711no—
247C2×C4⋊C4—2711no—
248C2×C4⋊C4C2×D27111yescharacteristic
249C2×C22⋊C4—2711no—
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⋊C4C2×D27111yescharacteristic
259C2×C22⋊C4C2×D27111yescharacteristic
260C2×C22⋊C4C2×D27111yescharacteristic
261C2×C42—2711no—
262C2×C42—2711no—
263#95—2711no—
264#95—2711no—
265#95—2711no—
266#95—2711no—
267#95—2711no—
268#95—2711no—
269#95—2711no—
270#95C2×D27111yescharacteristic
271#95C2×D27111yescharacteristic
272#95C2×D27111yescharacteristic
273C24—2711no—
274C24—2711no—
275C24C22×D27111yescharacteristic
276C2×D4—2714no—
277C2×D4—5422no—
278C2×D4—5422no—
279C2×D4—5422no—
280C2×D4—5422no—
281C2×D4—5422no—
282C2×D4—5422no—
283C2×D4—10841no—
284C2×D4—10841no—
285C22×C4—2711no—
286C22×C4—2711no—
287C22×C4—2711no—
288C22×C4—2711no—
289C22×C4—2711no—
290C22×C4—2711no—
291C22×C4—5422no—
292C22×C4—5422no—
293C22×C4—5422no—
294C22×C4—5422no—
295C22×C4—5422no—
296C22×C4—5422no—
297C22×C4—5422no—
298C22×C4—10841no—
299C22×C4—10841no—
300C22×C4C22×D27111yescharacteristic
301C22×C4C22×D27111yescharacteristic
302C22×C4C22×D27111yescharacteristic
303C22×C4C22×D27111yescharacteristic
304C22×C4C22×D27111yescharacteristic
305C22×C4C22×D27111yescharacteristic
306C4⋊C4—41no—
307C4⋊C4—5422no—
308C4⋊C4—5422no—
309C4⋊C4—5422no—
310C4⋊C4—10841no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—5422no—
315C22⋊C4—5422no—
316C22⋊C4—5422no—
317C22⋊C4—5422no—
318C22⋊C4—10841no—
319C22⋊C4—10841no—
320C22⋊C4—10841no—
321C22⋊C4—10841no—
322C22⋊C4—10841no—
323C42—5422no—
324C42—5422no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—2712no—
332C23—2712no—
333C23—5421no—
334C23—5421no—
335C23—5421no—
336C23—5421no—
337C23—5421no—
338C23—5421no—
339C23—5421no—
340C23—5421no—
341C23—5421no—
342C23—5421no—
343C23—5421no—
344C23—5421no—
345C23D4×D27112yes—
346C23C23×D27111yescharacteristic
347D4—5428no—
348D4—10844no—
349D4—10844no—
350C2×C4—21no—
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—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—2712no—
369C2×C4—2712no—
370C2×C4—2712no—
371C2×C4—2712no—
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—10844no—
387C2×C4—10844no—
388C2×C4—10844no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—5422no—
397C22—5422no—
398C22—5422no—
399C22—5422no—
400C22—5422no—
401C22—5422no—
402C22—10841no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—10841no—
410C22C2×D4×D27111yescharacteristic
411C22#41×D27111yescharacteristic
412C22#41×D27111yescharacteristic
413C22#41×D27111yescharacteristic
414C22#41×D27111yescharacteristic
415C22#2257620511yescharacteristic
416C22#2257620611yescharacteristic
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—2714no—
424C4—5422no—
425C4—10841no—
426C4—10841no—
427C4—10841no—
428C4—10841no—
429C2—42no—
430C2—10842no—
431C2—10842no—
432C2#123×D27111yescharacteristic
433C2#3177443011yescharacteristic
434C2#3177449911yescharacteristic
435C2#3177453611yescharacteristic
436C2#3177453911yescharacteristic
437C2#3177459811yescharacteristic
438C2#3177460911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#397×C271C2
prime radicalC2×#22576133C2
hypercenter#397D271
hypocenterC271#2420
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D271)⋊C4C2semidirect
3. #31773826C2semidirect
4. #31773814C2semidirect
5. #31773812C2semidirect
6. #31773797C2semidirect
7. #31773792C2semidirect
8. #31773775C2semidirect
9. #31773774C2semidirect
10. #31773761C2semidirect
11. #31773759C2semidirect
12. #31773758C2semidirect
13. #31773756C2semidirect
14. #31773744C2semidirect
15. #397×C271C2semidirect
219. C271#2420semidirect
224. #397D271semidirect

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
#317738262
#317738142
#317738122
#317737972
#317737922
#317737752
#317737742
#317737612
#317737592
#317737582
#317737562
#317737442
#397×C2712
#2420271