Finite Group Databasebeta

← #30670287

Subgroups — #30670287

Order 16768.

546 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862229C211yesmaximal, characteristic
2C2×#16862225C211yesmaximal, characteristic, complemented
3#22553254C211yesmaximal, characteristic, complemented
4#22553253C211yesmaximal, characteristic, complemented
5#22553241C211yesmaximal, characteristic, complemented
6#22553225C211yesmaximal, characteristic, complemented
7#22553216C211yesmaximal, characteristic, complemented
8#22553204C211yesmaximal, characteristic, complemented
9#22553201C211yesmaximal, characteristic, complemented
10#22553191C211yesmaximal, characteristic
11#22553191C211yesmaximal, characteristic
12#22553176C211yesmaximal, characteristic, complemented
13#22553175C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic131C211yesmaximal, characteristic, complemented
15C2×#95×C131C211yesmaximal, characteristic, complemented
16C22×Dic262C2211yescharacteristic
17C22×C22⋊D131C2211yescharacteristic
18#16862229—24no—
19#16862225—24no—
20C2×C4⋊Dic131C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×C22⋊Dic131C2211yescharacteristic
26C2×C22⋊Dic131C2211yescharacteristic
27C2×C22⋊Dic131C2211yescharacteristic
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×(C2×D131)⋊C4C2211yescharacteristic
33C2×(C2×D131)⋊C4C2211yescharacteristic
34C2×C4×Dic131C2211yescharacteristic
35C2×C4×Dic131C2211yescharacteristic
36C131⋊#95C2211yescharacteristic
37C131⋊#95C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C23×C4×C131C2211yescharacteristic
47C23×C4×C131C2211yescharacteristic
48C23×C4×C131C2211yescharacteristic
49#95×C131C2211yescharacteristic
50#95×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—
60C22×Dic131C2311yescharacteristic
61C22×Dic131C2311yescharacteristic
62C22×Dic131C2311yescharacteristic
63C22×Dic131C2311yescharacteristic
64C22×Dic131C2311yescharacteristic
65C22×Dic131C2311yescharacteristic
66C22×Dic131C2311yescharacteristic
67C4⋊Dic131—22no—
68Dic131⋊C4—22no—
69Dic131⋊C4—22no—
70Dic131⋊C4—22no—
71Dic131⋊C4—22no—
72C22⋊Dic131—22no—
73C22⋊Dic131—22no—
74C22⋊Dic131D414yes—
75(C2×D131)⋊C4—22no—
76(C2×D131)⋊C4—22no—
77(C2×D131)⋊C4—22no—
78(C2×D131)⋊C4—22no—
79(C2×D131)⋊C4—41no—
80(C2×D131)⋊C4—41no—
81C4×Dic131—22no—
82C4×Dic131—41no—
83C24×C131C2311yescharacteristic
84C22×C4×C131—22no—
85C22×C4×C131—22no—
86C22×C4×C131—22no—
87C22×C4×C131—22no—
88C22×C4×C131—22no—
89C22×C4×C131—22no—
90C22×C4×C131—22no—
91C22×C4×C131—41no—
92C22×C4×C131—41no—
93C22×C4×C131C2311yescharacteristic
94C22×C4×C131C2311yescharacteristic
95C22×C4×C131C2311yescharacteristic
96C22×C4×C131C2311yescharacteristic
97C22×C4×C131C2311yescharacteristic
98C22×C4×C131C2311yescharacteristic
99C22×D131—21no—
100C22×D131—21no—
101C22×D131—21no—
102C22×D131—21no—
103C22×D131—21no—
104C22×D131—21no—
105C22×D131—21no—
106Dic262—44no—
107C22⋊D131—44no—
108C2×Dic131—21no—
109C2×Dic131—21no—
110C2×Dic131—21no—
111C2×Dic131—21no—
112C2×Dic131—21no—
113C2×Dic131—21no—
114C2×Dic131—21no—
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×Dic131C2×D412yes—
128C2×Dic131C2×D412yes—
129C23×C131—21no—
130C23×C131—21no—
131C23×C131—21no—
132C23×C131—21no—
133C23×C131C2×D412yes—
134C23×C131#4112yes—
135C23×C131#4112yes—
136C23×C131C2411yescharacteristic
137C2×C4×C131—21no—
138C2×C4×C131—21no—
139C2×C4×C131—21no—
140C2×C4×C131—21no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131—21no—
151C2×C4×C131—21no—
152C2×C4×C131—21no—
153C2×C4×C131—44no—
154C2×C4×C131—44no—
155C2×C4×C131—44no—
156C2×C4×C131#4112yes—
157C2×C4×C131#4112yes—
158C22×C131—22no—
159C22×C131—22no—
160C22×C131—22no—
161C22×C131—22no—
162C22×C131—22no—
163C22×C131—22no—
164C22×C131#12414yes—
165C22×C131C22×D411yescharacteristic
166C22×C131C2×#4111yescharacteristic
167C22×C131C2×#4111yescharacteristic
168C22×C131C2×#4111yescharacteristic
169C22×C131C2×#4111yescharacteristic
170C22×C131H(5,𝔽2)11yescharacteristic
171C22×C1312−1+411yescharacteristic
172C2×D131—41no—
173C2×D131—41no—
174C2×D131—41no—
175C2×D131—41no—
176C2×D131—41no—
177C2×D131—41no—
178C2×D131—41no—
179C4×C131—22no—
180C4×C131—22no—
181C4×C131—41no—
182C4×C131—41no—
183C4×C131—41no—
184C4×C131—41no—
185Dic131—22no—
186Dic131—22no—
187Dic131—41no—
188Dic131—41no—
189Dic131—41no—
190Dic131—41no—
191Dic131—41no—
192C2×C131—24no—
193C2×C131C2×#12411yescharacteristic
194C2×C131#52911yescharacteristic
195C2×C131#53911yescharacteristic
196C2×C131#54611yescharacteristic
197C2×C131#54711yescharacteristic
198C2×C131#55111yescharacteristic
199C2×C131#55211yescharacteristic
200D131—81no—
201C131#248811yescharacteristic, complemented, Sylow
202#2488—1311nomaximal, Sylow
203C2×#123—1311no—
204C2×C22⋊Q8—1311no—
205#400—1311no—
206#400—1311no—
207#397—1311no—
208#393—1311no—
209#391—1311no—
210#388—1311no—
211#388—1311no—
212#386—1311no—
213#386—1311no—
214#382—1311no—
215#381—1311no—
216C4×C22⋊C4—1311no—
217C2×#95D13111yescharacteristic, complemented
218C22×Q8—1311no—
219C22×D4—1311no—
220C23×C4C2×D13111yescharacteristic
221C23×C4C2×D13111yescharacteristic
222C23×C4C2×D13111yescharacteristic
223#123—2624no—
224C22⋊Q8—2624no—
225C2×C4⋊C4—1311no—
226C2×C4⋊C4—1311no—
227C2×C4⋊C4—1311no—
228C2×C4⋊C4—1311no—
229C2×C4⋊C4—1311no—
230C2×C22⋊C4—1311no—
231C2×C22⋊C4—1311no—
232C2×C22⋊C4—1311no—
233C2×C22⋊C4—1311no—
234C2×C22⋊C4—1311no—
235C2×C22⋊C4—1311no—
236C2×C22⋊C4—1311no—
237C2×C22⋊C4—1311no—
238C2×C22⋊C4—1311no—
239C2×C42—1311no—
240C2×C42—1311no—
241#95—1311no—
242#95—1311no—
243#95—1311no—
244#95—1311no—
245#95—1311no—
246#95—1311no—
247#95—1311no—
248#95—1311no—
249#95—1311no—
250#95—1311no—
251#95C2×D13111yescharacteristic
252#95C2×D13111yescharacteristic
253#95C2×D13111yescharacteristic
254#95C2×D13111yescharacteristic
255C24—1311no—
256C24C22×D13111yescharacteristic
257C2×Q8—2622no—
258C2×Q8—2622no—
259C2×Q8—5241no—
260C2×D4—2622no—
261C2×D4—2622no—
262C2×D4—2622no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—1311no—
273C22×C4—1311no—
274C22×C4—1311no—
275C22×C4—1311no—
276C22×C4—1311no—
277C22×C4—1311no—
278C22×C4—1311no—
279C22×C4C22×D13111yescharacteristic
280C22×C4C22×D13111yescharacteristic
281C22×C4C22×D13111yescharacteristic
282C22×C4C22×D13111yescharacteristic
283C22×C4C22×D13111yescharacteristic
284C22×C4C22×D13111yescharacteristic
285C4⋊C4—2622no—
286C4⋊C4—2622no—
287C4⋊C4—2622no—
288C4⋊C4—2622no—
289C4⋊C4—2622no—
290C22⋊C4—2622no—
291C22⋊C4—2622no—
292C22⋊C4—2622no—
293C22⋊C4—2622no—
294C22⋊C4—2622no—
295C22⋊C4—2622no—
296C22⋊C4—5241no—
297C22⋊C4—5241no—
298C22⋊C4—1314no—
299C42—2622no—
300C42—5241no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—2621no—
306C23—2621no—
307C23—2621no—
308C23—2621no—
309C23—2621no—
310C23—2621no—
311C23—2621no—
312C23D4×D13112yes—
313C23#1452351212yes—
314C23#1452351312yes—
315C23C23×D13111yescharacteristic
316Q8—5244no—
317D4—5244no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—2621no—
338C2×C4—2621no—
339C2×C4—2621no—
340C2×C4—2621no—
341C2×C4—2621no—
342C2×C4—2621no—
343C2×C4—2621no—
344C2×C4—2621no—
345C2×C4—2621no—
346C2×C4—2621no—
347C2×C4—2621no—
348C2×C4—2621no—
349C2×C4—2621no—
350C2×C4—2621no—
351C2×C4—2621no—
352C2×C4—2621no—
353C2×C4—2621no—
354C2×C4—2621no—
355C2×C4—2621no—
356C2×C4—1312no—
357C2×C4—1312no—
358C2×C4#1452351312yes—
359C2×C4#1452351312yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—5241no—
367C22—5241no—
368C22—5241no—
369C22—5241no—
370C22—5241no—
371C22—5241no—
372C22—5241no—
373C22#1686223214yes—
374C22C2×D4×D13111yescharacteristic
375C22C2×#1452351211yescharacteristic
376C22C2×#1452351311yescharacteristic
377C22C2×#1452351311yescharacteristic
378C22C2×#1452351311yescharacteristic
379C22#1686229111yescharacteristic
380C22#1686229411yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—2622no—
388C4—2622no—
389C4—5241no—
390C4—5241no—
391C4—5241no—
392C4—5241no—
393C4—5241no—
394C2—24no—
395C2—10481no—
396C2C2×#1686223211yescharacteristic
397C2#2255384611yescharacteristic
398C2#2255390911yescharacteristic
399C2#2255395211yescharacteristic
400C2#2255396311yescharacteristic
401C2#2255398811yescharacteristic
402C2#2255399711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#95×C131C2
prime radicalC22×C22⋊D131C22
hypercenterC2×#95D131
hypocenterC131#2488
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. C2×#16862225C2semidirect
3. #22553254C2semidirect
4. #22553253C2semidirect
5. #22553241C2semidirect
6. #22553225C2semidirect
7. #22553216C2semidirect
8. #22553204C2semidirect
9. #22553201C2semidirect
12. #22553176C2semidirect
13. #22553175C2semidirect
14. C4×C22⋊Dic131C2semidirect
15. C2×#95×C131C2semidirect
201. C131#2488semidirect
217. C2×#95D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622292
C2×#168622252
#225532542
#225532532
#225532412
#225532252
#225532162
#225532042
#225532012
#225531912
#225531762
#225531752
C4×C22⋊Dic1312
C2×#95×C1312
#2488131