Finite Group Databasebeta

← #30671059

Subgroups — #30671059

Order 16768.

582 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×#16862233C211yesmaximal, characteristic, complemented
2C2×#16862228C211yesmaximal, characteristic, complemented
3#22553240C211yesmaximal, characteristic, complemented
4#22553240C211yesmaximal, characteristic, complemented
5#22553225C211yesmaximal, characteristic, complemented
6#22553220C211yesmaximal, characteristic
7#22553216C211yesmaximal, characteristic, complemented
8#22553212C211yesmaximal, characteristic, complemented
9#22553205C211yesmaximal, characteristic, complemented
10#22553199C211yesmaximal, characteristic, complemented
11#22553191C211yesmaximal, characteristic, complemented
12#22553188C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic131C211yesmaximal, characteristic, complemented
14#22553159C211yesmaximal, characteristic, complemented
15#386×C131C211yesmaximal, characteristic, complemented
16C22×Dic262C2211yescharacteristic
17C22×C22⋊D131C2211yescharacteristic
18C22×C22⋊D131C2211yescharacteristic
19C22×C4×D131C2211yescharacteristic
20#16862233—24no—
21#16862228—24no—
22C2×C4⋊Dic131C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×C22⋊Dic131C2211yescharacteristic
26C2×C22⋊Dic131C2211yescharacteristic
27C2×C22⋊Dic131C2211yescharacteristic
28C2×C22⋊Dic131C2211yescharacteristic
29C2×C22⋊Dic131C2211yescharacteristic
30C2×C22⋊Dic131C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×(C2×D131)⋊C4C2211yescharacteristic
33C2×(C2×D131)⋊C4C2211yescharacteristic
34C2×(C2×D131)⋊C4C2211yescharacteristic
35C2×C4×Dic131C2211yescharacteristic
36C2×C4×Dic131C2211yescharacteristic
37C2×C4×Dic131C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C22×D4×C131C2211yescharacteristic
47C23×C4×C131C2211yescharacteristic
48C2×C22⋊C4×C131C2211yescharacteristic
49C2×C22⋊C4×C131C2211yescharacteristic
50C2×C22⋊C4×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52#95×C131C2211yescharacteristic
53C23×D131C2311yescharacteristic
54C2×Dic262—22no—
55C2×Dic262—22no—
56C2×Dic262—41no—
57C2×C22⋊D131—22no—
58C2×C22⋊D131—22no—
59C2×C22⋊D131—22no—
60C2×C22⋊D131—22no—
61C2×C22⋊D131—41no—
62C2×C22⋊D131—41no—
63C22×Dic131C2311yescharacteristic
64C22×Dic131C2311yescharacteristic
65C22×Dic131C2311yescharacteristic
66C22×Dic131C2311yescharacteristic
67C22×Dic131C2311yescharacteristic
68C22×Dic131C2311yescharacteristic
69C22×Dic131C2311yescharacteristic
70C2×C4×D131—22no—
71C2×C4×D131—22no—
72C2×C4×D131—41no—
73C4⋊Dic131—22no—
74Dic131⋊C4—22no—
75Dic131⋊C4—22no—
76C22⋊Dic131—22no—
77C22⋊Dic131—22no—
78C22⋊Dic131—22no—
79C22⋊Dic131—22no—
80C22⋊Dic131—41no—
81C22⋊Dic131D414yescomplemented
82(C2×D131)⋊C4—22no—
83(C2×D131)⋊C4—22no—
84(C2×D131)⋊C4—22no—
85(C2×D131)⋊C4—41no—
86C4×Dic131—22no—
87C4×Dic131—41no—
88C4×Dic131—41no—
89C24×C131C2311yescharacteristic
90C24×C131C2311yescharacteristic
91C2×D4×C131—22no—
92C2×D4×C131—22no—
93C2×D4×C131—41no—
94C22×C4×C131—22no—
95C22×C4×C131—41no—
96C22×C4×C131—41no—
97C22×C4×C131C2311yescharacteristic
98C22×C4×C131C2311yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C22×C4×C131C2311yescharacteristic
101C22×C4×C131C2311yescharacteristic
102C22⋊C4×C131—22no—
103C22⋊C4×C131—22no—
104C22⋊C4×C131—41no—
105C22×D131—21no—
106C22×D131—21no—
107C22×D131—21no—
108C22×D131—21no—
109C22×D131—21no—
110C22×D131—21no—
111C22×D131#4112yes—
112Dic262—44no—
113C22⋊D131—44no—
114C22⋊D131—44no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131—21no—
118C2×Dic131—21no—
119C2×Dic131—21no—
120C2×Dic131—21no—
121C2×Dic131—21no—
122C2×Dic131—21no—
123C2×Dic131—21no—
124C2×Dic131—21no—
125C2×Dic131—21no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131—21no—
129C2×Dic131—21no—
130C2×Dic131—21no—
131C2×Dic131—21no—
132C2×Dic131—21no—
133C2×Dic131C2×D412yes—
134C2×Dic131C2×D412yes—
135C2×Dic131#4112yes—
136C4×D131—44no—
137C23×C131—21no—
138C23×C131—21no—
139C23×C131—21no—
140C23×C131—21no—
141C23×C131—21no—
142C23×C131—21no—
143C23×C131—21no—
144C23×C131—21no—
145C23×C131—21no—
146C23×C131—21no—
147C23×C131—21no—
148C23×C131—21no—
149C23×C131—21no—
150C23×C131C2×D412yes—
151C23×C131C2411yescharacteristic
152D4×C131—44no—
153C2×C4×C131—21no—
154C2×C4×C131—21no—
155C2×C4×C131—21no—
156C2×C4×C131—21no—
157C2×C4×C131—21no—
158C2×C4×C131—21no—
159C2×C4×C131—21no—
160C2×C4×C131—21no—
161C2×C4×C131—21no—
162C2×C4×C131—21no—
163C2×C4×C131—21no—
164C2×C4×C131—21no—
165C2×C4×C131—21no—
166C2×C4×C131—44no—
167C2×C4×C131#4112yes—
168C2×C4×C131#4112yes—
169C22×C131—22no—
170C22×C131—22no—
171C22×C131—22no—
172C22×C131—41no—
173C22×C131—41no—
174C22×C131—41no—
175C22×C131—41no—
176C22×C131—41no—
177C22×C131—41no—
178C22×C131—41no—
179C22×C131—41no—
180C22×C131—41no—
181C22×C131—41no—
182C22×C131—41no—
183C22×C131C22×D411yescharacteristic
184C22×C131C2×#4111yescharacteristic
185C22×C131C2×#4111yescharacteristic
186C22×C131C2×#4111yescharacteristic
187C22×C131C2×#4111yescharacteristic
188C22×C131H(5,𝔽2)11yescharacteristic
189C22×C131H(5,𝔽2)11yescharacteristic
190C2×D131—22no—
191C2×D131—22no—
192C2×D131—22no—
193C2×D131—41no—
194C2×D131—41no—
195C2×D131—41no—
196C2×D131—41no—
197C4×C131—22no—
198C4×C131—22no—
199C4×C131—41no—
200C4×C131—41no—
201C4×C131—41no—
202Dic131—22no—
203Dic131—22no—
204Dic131—22no—
205Dic131—41no—
206Dic131—41no—
207Dic131—41no—
208Dic131—41no—
209C2×C131—42no—
210C2×C131—81no—
211C2×C131#52911yescharacteristic
212C2×C131#53211yescharacteristic
213C2×C131#53811yescharacteristic
214C2×C131#54611yescharacteristic
215C2×C131#54611yescharacteristic
216C2×C131#55111yescharacteristic
217C2×C131#55911yescharacteristic
218D131—42no—
219C131#255711yescharacteristic, complemented, Sylow
220#2557—1311nomaximal, Sylow
221C2×#124—1311no—
222C2×#123—1311no—
223#397—1311no—
224#397—1311no—
225#393—1311no—
226#392—1311no—
227#391—1311no—
228#390—1311no—
229#388—1311no—
230#387—1311no—
231#386—1311no—
232#386—1311no—
233#386D13111yescharacteristic, complemented
234C4×C22⋊C4—1311no—
235C4×C22⋊C4—1311no—
236C22×Q8—1311no—
237C22×D4—1311no—
238C22×D4—1311no—
239C22×D4C2×D13111yescharacteristic
240C23×C4—1311no—
241C23×C4C2×D13111yescharacteristic
242#124—2624no—
243#123—2624no—
244C2×C4⋊C4—1311no—
245C2×C4⋊C4—1311no—
246C2×C4⋊C4—1311no—
247C2×C22⋊C4—1311no—
248C2×C22⋊C4—1311no—
249C2×C22⋊C4—1311no—
250C2×C22⋊C4—1311no—
251C2×C22⋊C4—1311no—
252C2×C22⋊C4—1311no—
253C2×C22⋊C4—1311no—
254C2×C22⋊C4—1311no—
255C2×C22⋊C4—1311no—
256C2×C22⋊C4—1311no—
257C2×C22⋊C4C2×D13111yescharacteristic
258C2×C22⋊C4C2×D13111yescharacteristic
259C2×C22⋊C4C2×D13111yescharacteristic
260C2×C42—1311no—
261C2×C42—1311no—
262C2×C42—1311no—
263#95—1311no—
264#95—1311no—
265#95—1311no—
266#95—1311no—
267#95—1311no—
268#95—1311no—
269#95—1311no—
270#95—1311no—
271#95C2×D13111yescharacteristic
272#95C2×D13111yescharacteristic
273C24—1311no—
274C24C22×D13111yescharacteristic
275C24C22×D13111yescharacteristic
276C2×Q8—2622no—
277C2×Q8—2622no—
278C2×Q8—5241no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—2622no—
283C2×D4—2622no—
284C2×D4—2622no—
285C2×D4—2622no—
286C2×D4—5241no—
287C2×D4—5241no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—2622no—
292C22×C4—2622no—
293C22×C4—5241no—
294C22×C4—1311no—
295C22×C4—1311no—
296C22×C4—1311no—
297C22×C4—1311no—
298C22×C4—1311no—
299C22×C4—1311no—
300C22×C4—1311no—
301C22×C4C22×D13111yescharacteristic
302C22×C4C22×D13111yescharacteristic
303C22×C4C22×D13111yescharacteristic
304C22×C4C22×D13111yescharacteristic
305C22×C4C22×D13111yescharacteristic
306C4⋊C4—2622no—
307C4⋊C4—2622no—
308C4⋊C4—2622no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—2622no—
313C22⋊C4—2622no—
314C22⋊C4—2622no—
315C22⋊C4—2622no—
316C22⋊C4—2622no—
317C22⋊C4—2622no—
318C22⋊C4—2622no—
319C22⋊C4—5241no—
320C22⋊C4—5241no—
321C22⋊C4—1314no—
322C42—2622no—
323C42—5241no—
324C42—5241no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—2621no—
339C23—2621no—
340C23—2621no—
341C23—2621no—
342C23—2621no—
343C23—2621no—
344C23—1312no—
345C23D4×D13112yes—
346C23C23×D13111yescharacteristic
347Q8—5244no—
348D4—44no—
349D4—5244no—
350D4—5244no—
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—2621no—
366C2×C4—2621no—
367C2×C4—2621no—
368C2×C4—2621no—
369C2×C4—2621no—
370C2×C4—2621no—
371C2×C4—2621no—
372C2×C4—2621no—
373C2×C4—2621no—
374C2×C4—2621no—
375C2×C4—2621no—
376C2×C4—2621no—
377C2×C4—2621no—
378C2×C4—2621no—
379C2×C4—2621no—
380C2×C4—2621no—
381C2×C4—2621no—
382C2×C4—2621no—
383C2×C4—5244no—
384C2×C4—1312no—
385C2×C4—1312no—
386C2×C4—1312no—
387C2×C4#1452351212yes—
388C2×C4#1452351312yes—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2622no—
404C22—2622no—
405C22—2622no—
406C22—5241no—
407C22—5241no—
408C22—5241no—
409C22—5241no—
410C22C2×D4×D13111yescharacteristic
411C22#41×D13111yescharacteristic
412C22#41×D13111yescharacteristic
413C22C2×#1452351211yescharacteristic
414C22C2×#1452351311yescharacteristic
415C22#1686229111yescharacteristic
416C22#1686229111yescharacteristic
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—2622no—
423C4—2622no—
424C4—2622no—
425C4—5241no—
426C4—5241no—
427C4—5241no—
428C4—5241no—
429C2—42no—
430C2—81no—
431C2—5242no—
432C2#2255384411yescharacteristic
433C2#2255385711yescharacteristic
434C2#2255389711yescharacteristic
435C2#2255395111yescharacteristic
436C2#2255395211yescharacteristic
437C2#2255399011yescharacteristic
438C2#2255404511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#386×C131C2
prime radical#22553220C2
hypercenter#386D131
hypocenterC131#2557
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#16862233C2semidirect
2. C2×#16862228C2semidirect
3. #22553240C2semidirect
4. #22553240C2semidirect
5. #22553225C2semidirect
7. #22553216C2semidirect
8. #22553212C2semidirect
9. #22553205C2semidirect
10. #22553199C2semidirect
11. #22553191C2semidirect
12. #22553188C2semidirect
13. C4×C22⋊Dic131C2semidirect
14. #22553159C2semidirect
15. #386×C131C2semidirect
81. C22⋊Dic131D4semidirect
219. C131#2557semidirect
233. #386D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622332
C2×#168622282
#225532402
#225532252
#225532202
#225532162
#225532122
#225532052
#225531992
#225531912
#225531882
C4×C22⋊Dic1312
#225531592
#386×C1312
#2557131