Finite Group Databasebeta

← #46839775

Subgroups — #46839775

Order 34688.

704 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576132C211yesmaximal, characteristic, complemented
2C2×#22576132C211yesmaximal, characteristic, complemented
3C2×D4×Dic271C211yesmaximal, characteristic, complemented
4C2×#22576116C211yesmaximal, characteristic, complemented
5#31773813C211yesmaximal, characteristic, complemented
6#31773811C211yesmaximal, characteristic, complemented
7#31773805C211yesmaximal, characteristic, complemented
8#31773800C211yesmaximal, characteristic, complemented
9#31773797C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773794C211yesmaximal, characteristic, complemented
12#31773790C211yesmaximal, characteristic, complemented
13#31773789C211yesmaximal, characteristic, complemented
14#31773755C211yesmaximal, characteristic, complemented
15#392×C271C211yesmaximal, characteristic, complemented
16C22×C22⋊D271C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C23×Dic271C2211yescharacteristic
20C23×Dic271C2211yescharacteristic
21#22576132—24no—
22#22576132—24no—
23D4×Dic271—24no—
24#22576116—24no—
25C2×C4⋊Dic271C2211yescharacteristic
26C2×C4⋊Dic271C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C22⋊Dic271C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×C4×Dic271C2211yescharacteristic
43C2×C4×Dic271C2211yescharacteristic
44C2×C4×Dic271C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C22×D4×C271C2211yescharacteristic
49C22×D4×C271C2211yescharacteristic
50C22×D4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54#95×C271C2211yescharacteristic
55C23×D271C2311yescharacteristic
56C2×C22⋊D271—22no—
57C2×C22⋊D271—22no—
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—22no—
62C2×C22⋊D271—41no—
63C2×C22⋊D271—41no—
64C2×C22⋊D271—41no—
65C22×Dic271—22no—
66C22×Dic271—22no—
67C22×Dic271—22no—
68C22×Dic271—22no—
69C22×Dic271—22no—
70C22×Dic271D414yescomplemented
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C22×Dic271C2311yescharacteristic
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2311yescharacteristic
76C22×Dic271C2311yescharacteristic
77C22×Dic271C2311yescharacteristic
78C4⋊Dic271—22no—
79C4⋊Dic271—41no—
80Dic271⋊C4—22no—
81Dic271⋊C4—22no—
82Dic271⋊C4—41no—
83Dic271⋊C4D414yescomplemented
84C22⋊Dic271—22no—
85C22⋊Dic271—22no—
86C22⋊Dic271—22no—
87C22⋊Dic271—22no—
88C22⋊Dic271—22no—
89C22⋊Dic271—41no—
90C22⋊Dic271—41no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—22no—
94(C2×D271)⋊C4—41no—
95C4×Dic271—22no—
96C4×Dic271—22no—
97C4×Dic271—22no—
98C24×C271C2311yescharacteristic
99C24×C271C2311yescharacteristic
100C24×C271C2311yescharacteristic
101C2×D4×C271—22no—
102C2×D4×C271—22no—
103C2×D4×C271—22no—
104C2×D4×C271—22no—
105C2×D4×C271—22no—
106C2×D4×C271—22no—
107C2×D4×C271—41no—
108C2×D4×C271—41no—
109C2×D4×C271—41no—
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114C22⋊C4×C271—22no—
115C22⋊C4×C271—22no—
116C22⋊C4×C271—41no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—21no—
123C22×D271—21no—
124C22⋊D271—44no—
125C22⋊D271—44no—
126C22⋊D271—44no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—28no—
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×Dic271—21no—
144C2×Dic271—44no—
145C2×Dic271C2×D412yes—
146C2×Dic271C2×D412yes—
147C2×Dic271C2×D412yes—
148C2×Dic271C2×D412yes—
149C2×Dic271#4112yes—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—21no—
153C23×C271—21no—
154C23×C271—21no—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271—21no—
158C23×C271—21no—
159C23×C271—21no—
160C23×C271—21no—
161C23×C271—21no—
162C23×C271—21no—
163C23×C271—21no—
164C23×C271—21no—
165C23×C271—21no—
166C23×C271—21no—
167C23×C271—21no—
168C23×C271—21no—
169C23×C271C2×D412yes—
170C23×C271#4112yes—
171C23×C271C2411yescharacteristic
172D4×C271—28no—
173D4×C271—44no—
174D4×C271—44no—
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×C271—21no—
182C2×C4×C271—21no—
183C2×C4×C271—21no—
184C2×C4×C271—21no—
185C2×C4×C271C2×D412yes—
186C2×C4×C271#4112yes—
187C22×C271—22no—
188C22×C271—22no—
189C22×C271—22no—
190C22×C271—22no—
191C22×C271—22no—
192C22×C271—22no—
193C22×C271—41no—
194C22×C271—41no—
195C22×C271—41no—
196C22×C271—41no—
197C22×C271—41no—
198C22×C271—41no—
199C22×C271—41no—
200C22×C271—41no—
201C22×C271—41no—
202C22×C271—41no—
203C22×C271—41no—
204C22×C271—41no—
205C22×C271—41no—
206C22×C271—41no—
207C22×C271—41no—
208C22×C271C22×D411yescharacteristic
209C22×C271C22×D411yescharacteristic
210C22×C271C2×#4111yescharacteristic
211C22×C271C2×#4111yescharacteristic
212C22×C271C2×#4111yescharacteristic
213C22×C271H(5,𝔽2)11yescharacteristic
214C22×C271H(5,𝔽2)11yescharacteristic
215C2×D271—41no—
216C2×D271—41no—
217C2×D271—41no—
218C2×D271—41no—
219C2×D271—41no—
220C2×D271—41no—
221C2×D271—41no—
222C4×C271—22no—
223C4×C271—22no—
224C4×C271—41no—
225C4×C271—41no—
226Dic271—22no—
227Dic271—22no—
228Dic271—41no—
229Dic271—41no—
230Dic271—41no—
231Dic271—41no—
232Dic271#12114yes—
233C2×C271—42no—
234C2×C271—42no—
235C2×C271—81no—
236C2×C271C2×#12111yescharacteristic
237C2×C271#52511yescharacteristic
238C2×C271#53511yescharacteristic
239C2×C271#54611yescharacteristic
240C2×C271#55011yescharacteristic
241C2×C271#55311yescharacteristic
242C2×C271#55511yescharacteristic
243D271—81no—
244C271#241411yescharacteristic, complemented, Sylow
245#2414—2711nomaximal, Sylow
246C2×#121—2711no—
247C2×#121—2711no—
248C2×C4×D4—2711no—
249C2×#117—2711no—
250#394—2711no—
251#394—2711no—
252#392—2711no—
253#392D27111yescharacteristic, complemented
254#390—2711no—
255#390—2711no—
256#390—2711no—
257#389—2711no—
258#388—2711no—
259#388—2711no—
260#380—2711no—
261C22×D4—2711no—
262C22×D4—2711no—
263C22×D4—2711no—
264C22×D4C2×D27111yescharacteristic
265C22×D4C2×D27111yescharacteristic
266C22×D4C2×D27111yescharacteristic
267C23×C4—2711no—
268C23×C4—2711no—
269#121—5424no—
270#121—5424no—
271C4×D4—5424no—
272#117—5424no—
273C2×C4⋊C4—2711no—
274C2×C4⋊C4—2711no—
275C2×C4⋊C4—2711no—
276C2×C4⋊C4—2711no—
277C2×C4⋊C4—2711no—
278C2×C4⋊C4—2711no—
279C2×C22⋊C4—2711no—
280C2×C22⋊C4—2711no—
281C2×C22⋊C4—2711no—
282C2×C22⋊C4—2711no—
283C2×C22⋊C4—2711no—
284C2×C22⋊C4—2711no—
285C2×C22⋊C4—2711no—
286C2×C22⋊C4—2711no—
287C2×C22⋊C4—2711no—
288C2×C22⋊C4—2711no—
289C2×C22⋊C4—2711no—
290C2×C22⋊C4C2×D27111yescharacteristic
291C2×C22⋊C4C2×D27111yescharacteristic
292C2×C22⋊C4C2×D27111yescharacteristic
293C2×C42—2711no—
294C2×C42—2711no—
295C2×C42—2711no—
296#95—2711no—
297#95—2711no—
298#95—2711no—
299#95C2×D27111yescharacteristic
300C24—2711no—
301C24C22×D27111yescharacteristic
302C24C22×D27111yescharacteristic
303C24C22×D27111yescharacteristic
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—5422no—
314C2×D4—5422no—
315C2×D4—5422no—
316C2×D4—5422no—
317C2×D4—5422no—
318C2×D4—5422no—
319C2×D4—10841no—
320C2×D4—10841no—
321C2×D4—10841no—
322C22×C4—2711no—
323C22×C4—2714no—
324C22×C4—2711no—
325C22×C4—2711no—
326C22×C4—2711no—
327C22×C4—2711no—
328C22×C4—2711no—
329C22×C4—2711no—
330C22×C4—5422no—
331C22×C4—5422no—
332C22×C4—5422no—
333C22×C4—5422no—
334C22×C4—5422no—
335C22×C4C22×D27111yescharacteristic
336C22×C4C22×D27111yescharacteristic
337C22×C4C22×D27111yescharacteristic
338C22×C4C22×D27111yescharacteristic
339C4⋊C4—2714no—
340C4⋊C4—5422no—
341C4⋊C4—5422no—
342C4⋊C4—5422no—
343C4⋊C4—10841no—
344C4⋊C4—10841no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—41no—
348C22⋊C4—5422no—
349C22⋊C4—5422no—
350C22⋊C4—5422no—
351C22⋊C4—5422no—
352C22⋊C4—5422no—
353C22⋊C4—5422no—
354C22⋊C4—5422no—
355C22⋊C4—5422no—
356C22⋊C4—10841no—
357C22⋊C4—10841no—
358C22⋊C4—10841no—
359C42—5422no—
360C42—5422no—
361C42—5422no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—5421no—
382C23—5421no—
383C23—5421no—
384C23—5421no—
385C23—5421no—
386C23—5421no—
387C23—5421no—
388C23D4×D27112yes—
389C23#1686465312yes—
390C23C23×D27111yescharacteristic
391D4—28no—
392D4—44no—
393D4—44no—
394D4—10844no—
395D4—10844no—
396D4—10844no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—2712no—
408C2×C4—2712no—
409C2×C4—2712no—
410C2×C4—2712no—
411C2×C4—2712no—
412C2×C4—5428no—
413C2×C4—5421no—
414C2×C4—5421no—
415C2×C4—5421no—
416C2×C4—5421no—
417C2×C4—5421no—
418C2×C4—5421no—
419C2×C4—5421no—
420C2×C4—5421no—
421C2×C4—5421no—
422C2×C4—5421no—
423C2×C4—5421no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4—5421no—
429C2×C4—10844no—
430C2×C4D4×D27112yes—
431C2×C4#1686465312yes—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—10841no—
454C22—10841no—
455C22—10841no—
456C22—10841no—
457C22—10841no—
458C22—10841no—
459C22—10841no—
460C22C2×D4×D27111yescharacteristic
461C22C2×D4×D27111yescharacteristic
462C22#41×D27111yescharacteristic
463C22C2×#1686465311yescharacteristic
464C22C2×#1686465311yescharacteristic
465C22#2257620511yescharacteristic
466C22#2257620511yescharacteristic
467C4—22no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—2714no—
472C4—5422no—
473C4—5422no—
474C4—10841no—
475C4—10841no—
476C4—10841no—
477C4—10841no—
478C2—42no—
479C2—42no—
480C2—81no—
481C2—21681no—
482C2#121×D27111yescharacteristic
483C2#3177440311yescharacteristic
484C2#3177446211yescharacteristic
485C2#3177453411yescharacteristic
486C2#3177456711yescharacteristic
487C2#3177459011yescharacteristic
488C2#3177461111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#392×C271C2
hypercenter#392D271
hypocenterC271#2414
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#22576132C2semidirect
2. C2×#22576132C2semidirect
3. C2×D4×Dic271C2semidirect
4. C2×#22576116C2semidirect
5. #31773813C2semidirect
6. #31773811C2semidirect
7. #31773805C2semidirect
8. #31773800C2semidirect
9. #31773797C2semidirect
10. #31773797C2semidirect
11. #31773794C2semidirect
12. #31773790C2semidirect
13. #31773789C2semidirect
14. #31773755C2semidirect
15. #392×C271C2semidirect
70. C22×Dic271D4semidirect
83. Dic271⋊C4D4semidirect
244. C271#2414semidirect
253. #392D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761322
C2×D4×Dic2712
C2×#225761162
#317738132
#317738112
#317738052
#317738002
#317737972
#317737942
#317737902
#317737892
#317737552
#392×C2712
#2414271