Finite Group Databasebeta

← #46839934

Subgroups — #46839934

Order 34688.

660 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×#22576146C211yesmaximal, characteristic, complemented
2C2×#22576137C211yesmaximal, characteristic, complemented
3C2×#22576132C211yesmaximal, characteristic
4C2×(C22⋊D271)⋊C4C211yesmaximal, characteristic, complemented
5#31773826C211yesmaximal, characteristic, complemented
6#31773822C211yesmaximal, characteristic, complemented
7#31773812C211yesmaximal, characteristic, complemented
8#31773801C211yesmaximal, characteristic, complemented
9#31773792C211yesmaximal, characteristic, complemented
10#31773777C211yesmaximal, characteristic, complemented
11#31773773C211yesmaximal, characteristic, complemented
12#31773755C211yesmaximal, characteristic, complemented
13#31773754C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic271C211yesmaximal, characteristic, complemented
15#394×C271C211yesmaximal, characteristic, complemented
16C22×Dic542C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C23×Dic271C2211yescharacteristic
20C23×Dic271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22#22576146—24no—
23#22576137—24no—
24#22576132—24no—
25(C22⋊D271)⋊C4—24no—
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×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×C4×Dic271C2211yescharacteristic
40C2×C4×Dic271C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C22×D4×C271C2211yescharacteristic
49C2×C4⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54#95×C271C2211yescharacteristic
55C23×D271C2311yescharacteristic
56C2×Dic542—22no—
57C2×Dic542—22no—
58C2×Dic542—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—22no—
62C2×C22⋊D271—22no—
63C2×C22⋊D271—41no—
64C2×C22⋊D271D414yes—
65C22×Dic271—22no—
66C22×Dic271—22no—
67C22×Dic271—22no—
68C22×Dic271—22no—
69C22×Dic271—22no—
70C22×Dic271—41no—
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C22×Dic271C2311yescharacteristic
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2311yescharacteristic
76C22×Dic271C2311yescharacteristic
77C22×Dic271C2311yescharacteristic
78C2×C4×D271—22no—
79C2×C4×D271—22no—
80C2×C4×D271—41no—
81C4⋊Dic271—22no—
82Dic271⋊C4—22no—
83Dic271⋊C4—22no—
84Dic271⋊C4—41no—
85Dic271⋊C4D414yescomplemented
86C22⋊Dic271—22no—
87C22⋊Dic271—22no—
88C22⋊Dic271—22no—
89C22⋊Dic271—22no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—22no—
94C4×Dic271—22no—
95C4×Dic271—22no—
96C24×C271C2311yescharacteristic
97C24×C271C2311yescharacteristic
98C2×D4×C271—22no—
99C2×D4×C271—22no—
100C2×D4×C271—41no—
101C22×C4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103C22×C4×C271C2311yescharacteristic
104C22×C4×C271C2311yescharacteristic
105C22×C4×C271C2311yescharacteristic
106C4⋊C4×C271—41no—
107C22⋊C4×C271—22no—
108C22⋊C4×C271—22no—
109C22⋊C4×C271—22no—
110C22⋊C4×C271—22no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—21no—
116C22×D271—21no—
117C22×D271C2×D412yes—
118Dic542—44no—
119C22⋊D271—28no—
120C22⋊D271—44no—
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—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—44no—
138C2×Dic271—44no—
139C2×Dic271C2×D412yes—
140C2×Dic271C2×D412yes—
141C2×Dic271C2×D412yes—
142C2×Dic271#4112yes—
143C2×Dic271#4112yes—
144C4×D271—44no—
145C23×C271—21no—
146C23×C271—21no—
147C23×C271—21no—
148C23×C271—21no—
149C23×C271—21no—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—21no—
153C23×C271—21no—
154C23×C271—21no—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271C2×D412yes—
158C23×C271#4112yes—
159C23×C271C2411yescharacteristic
160D4×C271—44no—
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—21no—
172C2×C4×C271—21no—
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271C2×D412yes—
176C22×C271—22no—
177C22×C271—22no—
178C22×C271—22no—
179C22×C271—22no—
180C22×C271—22no—
181C22×C271—22no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271C22×D411yescharacteristic
191C22×C271C22×D411yescharacteristic
192C22×C271C2×#4111yescharacteristic
193C22×C271C2×#4111yescharacteristic
194C22×C271C2×#4111yescharacteristic
195C22×C271H(5,𝔽2)11yescharacteristic
196C22×C2712−1+411yescharacteristic
197C2×D271—22no—
198C2×D271—22no—
199C2×D271—22no—
200C2×D271—41no—
201C2×D271—41no—
202C2×D271—41no—
203C2×D271—41no—
204C4×C271—22no—
205C4×C271—41no—
206C4×C271—41no—
207C4×C271—41no—
208C4×C271—41no—
209Dic271—22no—
210Dic271—22no—
211Dic271—41no—
212Dic271—41no—
213Dic271—41no—
214Dic271—41no—
215Dic271#12114yes—
216C2×C271—42no—
217C2×C271—42no—
218C2×C271C2×#12111yescharacteristic
219C2×C271#52911yescharacteristic
220C2×C271#54211yescharacteristic
221C2×C271#54611yescharacteristic
222C2×C271#54611yescharacteristic
223C2×C271#54711yescharacteristic
224C2×C271#54811yescharacteristic
225D271—42no—
226C271#242711yescharacteristic, complemented, Sylow
227#2427—2711nomaximal, Sylow
228C2×#124—2711no—
229C2×C22⋊Q8—2711no—
230C2×#121—2711no—
231C2×C4×D4—2711no—
232#397—2711no—
233#397—2711no—
234#394—2711no—
235#394D27111yescharacteristic, complemented
236#391—2711no—
237#389—2711no—
238#386—2711no—
239#386—2711no—
240#380—2711no—
241#380—2711no—
242C4×C22⋊C4—2711no—
243C22×Q8—2711no—
244C22×D4—2711no—
245C22×D4—2711no—
246C22×D4C2×D27111yescharacteristic
247C23×C4—2711no—
248C23×C4—2711no—
249C23×C4—2711no—
250#124—5424no—
251C22⋊Q8—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⋊C4C2×D27111yescharacteristic
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⋊C4—2711no—
267C2×C22⋊C4—2711no—
268C2×C22⋊C4C2×D27111yescharacteristic
269C2×C22⋊C4C2×D27111yescharacteristic
270C2×C22⋊C4C2×D27111yescharacteristic
271C2×C22⋊C4C2×D27111yescharacteristic
272C2×C42—2711no—
273C2×C42—2711no—
274#95—2711no—
275#95—2711no—
276#95—2711no—
277#95—2711no—
278#95—2711no—
279#95—2711no—
280#95—2711no—
281#95C2×D27111yescharacteristic
282C24—2711no—
283C24C22×D27111yescharacteristic
284C24C22×D27111yescharacteristic
285C2×Q8—5422no—
286C2×Q8—5422no—
287C2×Q8—5422no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—41no—
291C2×D4—2714no—
292C2×D4—5422no—
293C2×D4—5422no—
294C2×D4—5422no—
295C2×D4—5422no—
296C2×D4—10841no—
297C22×C4—2711no—
298C22×C4—2711no—
299C22×C4—2711no—
300C22×C4—2711no—
301C22×C4—2711no—
302C22×C4—2711no—
303C22×C4—2711no—
304C22×C4—5422no—
305C22×C4—5422no—
306C22×C4—5422no—
307C22×C4—5422no—
308C22×C4—5422no—
309C22×C4—5422no—
310C22×C4—5422no—
311C22×C4—10841no—
312C22×C4—10841no—
313C22×C4C22×D27111yescharacteristic
314C22×C4C22×D27111yescharacteristic
315C22×C4C22×D27111yescharacteristic
316C22×C4C22×D27111yescharacteristic
317C22×C4C22×D27111yescharacteristic
318C4⋊C4—41no—
319C4⋊C4—2714no—
320C4⋊C4—5422no—
321C4⋊C4—5422no—
322C4⋊C4—5422no—
323C4⋊C4—10841no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—5422no—
329C22⋊C4—5422no—
330C22⋊C4—5422no—
331C22⋊C4—5422no—
332C22⋊C4—5422no—
333C22⋊C4—5422no—
334C22⋊C4—5422no—
335C22⋊C4—5422no—
336C42—5422no—
337C42—5422no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—2712no—
351C23—5421no—
352C23—5421no—
353C23—5421no—
354C23—5421no—
355C23—5421no—
356C23—5421no—
357C23D4×D27112yes—
358C23#1686465312yes—
359C23C23×D27111yescharacteristic
360Q8—10844no—
361D4—44no—
362D4—5428no—
363D4—10844no—
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—2712no—
379C2×C4—2712no—
380C2×C4—2712no—
381C2×C4—2712no—
382C2×C4—2712no—
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—5421no—
395C2×C4—5421no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—5421no—
399C2×C4—10844no—
400C2×C4—10844no—
401C2×C4—10844no—
402C2×C4D4×D27112yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—5422no—
418C22—5422no—
419C22—5422no—
420C22—10841no—
421C22—10841no—
422C22—10841no—
423C22—10841no—
424C22C2×D4×D27111yescharacteristic
425C22C2×D4×D27111yescharacteristic
426C22#41×D27111yescharacteristic
427C22#41×D27111yescharacteristic
428C22C2×#1686465311yescharacteristic
429C22#2257620511yescharacteristic
430C22#2257620711yescharacteristic
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—2714no—
437C4—5422no—
438C4—5422no—
439C4—10841no—
440C4—10841no—
441C4—10841no—
442C4—10841no—
443C2—42no—
444C2—42no—
445C2—10842no—
446C2#121×D27111yescharacteristic
447C2#3177442411yescharacteristic
448C2#3177451111yescharacteristic
449C2#3177453411yescharacteristic
450C2#3177453611yescharacteristic
451C2#3177454511yescharacteristic
452C2#3177455311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#394×C271C2
prime radicalC2×#22576132C2
hypercenter#394D271
hypocenterC271#2427
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22576146C2semidirect
2. C2×#22576137C2semidirect
4. C2×(C22⋊D271)⋊C4C2semidirect
5. #31773826C2semidirect
6. #31773822C2semidirect
7. #31773812C2semidirect
8. #31773801C2semidirect
9. #31773792C2semidirect
10. #31773777C2semidirect
11. #31773773C2semidirect
12. #31773755C2semidirect
13. #31773754C2semidirect
14. C22⋊C4×Dic271C2semidirect
15. #394×C271C2semidirect
85. Dic271⋊C4D4semidirect
226. C271#2427semidirect
235. #394D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761462
C2×#225761372
C2×#225761322
C2×(C22⋊D271)⋊C42
#317738262
#317738222
#317738122
#317738012
#317737922
#317737772
#317737732
#317737552
#317737542
C22⋊C4×Dic2712
#394×C2712
#2427271