Finite Group Databasebeta

← #34304850

Subgroups — #34304850

Order 20224.

938 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11929608C212yesmaximal
2#23764394C214yesmaximal
3#23764390C212yesmaximal
4#23763396C211yesmaximal, characteristic
5#23763396C212yesmaximal
6C4×#14579080C211yesmaximal, characteristic
7C4×#14579080C212yesmaximal
8C2×#557×C79C211yesmaximal, characteristic
9#19098694—24no—
10#19098694—22no—
11Q8×C79⋊C8—22no—
12#19098587C2212yes—
13#19098587C2214yes—
14C2×#14579080C2211yescharacteristic
15C2×#14579080C2212yes—
16C2×#14579080C2211yescharacteristic
17C4×#11929608C2212yes—
18C4×#11929608C2214yes—
19#19098332C2214yes—
20#19098332C2214yes—
21#19098332C2212yes—
22#19098332C2212yes—
23#557×C79—22no—
24#557×C79—24no—
25#557×C79—22no—
26C2×C4⋊Q8×C79C412yes—
27C2×C4⋊Q8×C79C2211yescharacteristic
28C2×#125×C79C414yes—
29C2×#125×C79C2212yes—
30C2×C4×Q8×C79C412yes—
31C2×C4×Q8×C79C2212yes—
32C2×C4×Q8×C79C2211yescharacteristic
33C2×C4×Q8×C79C2211yescharacteristic
34C2×#11929608C2312yes—
35C2×#11929608C2314yes—
36C2×#11929608C2312yes—
37C4⋊(C79⋊C8)—24no—
38C4⋊(C79⋊C8)—22no—
39C4⋊(C79⋊C8)—24no—
40C4⋊(C79⋊C8)—22no—
41#14579080—22no—
42#14579080—24no—
43#14579080Q814yescomplemented
44#14579072—24no—
45#14579072—22no—
46C4×C79⋊C8—24no—
47C4×C79⋊C8—22no—
48C22×Q8×C79C2×C412yes—
49C22×Q8×C79C2×C411yescharacteristic
50C22×Q8×C79C2311yescharacteristic
51C4⋊Q8×C79—24no—
52C4⋊Q8×C79—24no—
53C4⋊Q8×C79—22no—
54C4⋊Q8×C79—22no—
55#125×C79—28no—
56#125×C79—28no—
57#125×C79—24no—
58#125×C79—24no—
59C4×Q8×C79—24no—
60C4×Q8×C79—22no—
61C4×Q8×C79—22no—
62C4×Q8×C79—24no—
63C4×Q8×C79—24no—
64C4×Q8×C79—22no—
65C4×Q8×C79—22no—
66C4×Q8×C79D412yes—
67C4×Q8×C79D414yes—
68C4×Q8×C79D412yes—
69C2×C4⋊C4×C79C2×C414yes—
70C2×C4⋊C4×C79C2×C412yes—
71C2×C4⋊C4×C79C2×C414yes—
72C2×C4⋊C4×C79C2×C412yes—
73C2×C4⋊C4×C79C2×C414yes—
74C2×C4⋊C4×C79C2×C411yescharacteristic
75C2×C4⋊C4×C79C2×C412yes—
76C2×C4⋊C4×C79C2311yescharacteristic
77C2×C4⋊C4×C79C2312yes—
78C2×C42×C79C2×C414yes—
79C2×C42×C79C2×C412yes—
80C2×C42×C79C2312yes—
81C2×C42×C79C2311yescharacteristic
82#11929608—22no—
83#11929608—24no—
84#11929608—22no—
85#11929608—22no—
86#11929608—24no—
87#11929608C2×Q814yes—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—24no—
90C2×C79⋊C8—22no—
91C2×Q8×C79—22no—
92C2×Q8×C79—22no—
93C2×Q8×C79—22no—
94C2×Q8×C79—22no—
95C2×Q8×C79—42no—
96C2×Q8×C79—44no—
97C2×Q8×C79C22⋊C412yes—
98C2×Q8×C79C22⋊C412yes—
99C2×Q8×C79C2×Q812yes—
100C2×Q8×C79#4112yes—
101C22×C4×C79C22×C411yescharacteristic
102C22×C4×C79C22×C412yes—
103C22×C4×C79C22×C412yes—
104C22×C4×C79C22×C414yes—
105C22×C4×C79C22×C412yes—
106C22×C4×C79C22×C411yescharacteristic
107C22×C4×C79C22×C412yes—
108C22×C4×C79C2411yescharacteristic
109C4⋊C4×C79—24no—
110C4⋊C4×C79—24no—
111C4⋊C4×C79—24no—
112C4⋊C4×C79—28no—
113C4⋊C4×C79—28no—
114C4⋊C4×C79—24no—
115C4⋊C4×C79—22no—
116C4⋊C4×C79—24no—
117C4⋊C4×C79C22⋊C412yes—
118C4⋊C4×C79C22⋊C411yescharacteristic
119C4⋊C4×C79C22⋊C411yescharacteristic
120C4⋊C4×C79C22⋊C414yes—
121C4⋊C4×C79C22⋊C414yes—
122C42×C79—24no—
123C42×C79—24no—
124C42×C79—22no—
125C42×C79—22no—
126C42×C79C2×D414yes—
127C42×C79C2×D414yes—
128C42×C79C2×D412yes—
129C42×C79C2×D412yes—
130C23×C79C23×C411yescharacteristic
131Q8×C79—24no—
132Q8×C79—24no—
133Q8×C79—24no—
134Q8×C79—42no—
135Q8×C79—44no—
136Q8×C79—44no—
137C2×C4×C79—22no—
138C2×C4×C79—22no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—24no—
142C2×C4×C79—42no—
143C2×C4×C79—41no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79—44no—
147C2×C4×C79C2×C22⋊C412yes—
148C2×C4×C79C2×C22⋊C412yes—
149C2×C4×C79C2×C22⋊C411yescharacteristic
150C2×C4×C79C2×C22⋊C412yes—
151C2×C4×C79C2×C22⋊C414yes—
152C2×C4×C79C2×C22⋊C411yescharacteristic
153C2×C4×C79C4×Q814yes—
154C2×C4×C79C22×D412yes—
155C2×C4×C79C22×Q811yescharacteristic
156C2×C4×C79C2×#4111yescharacteristic
157C2×C4×C792−1+412yes—
158C79⋊C8—24no—
159C79⋊C8—44no—
160C79⋊C8—42no—
161C22×C79—21no—
162C22×C79—21no—
163C22×C79C22×C22⋊C411yescharacteristic
164C22×C79C2×C4×Q811yescharacteristic
165C22×C79#51911yescharacteristic
166C4×C79—24no—
167C4×C79—21no—
168C4×C79—24no—
169C4×C79—22no—
170C4×C79—22no—
171C4×C79—22no—
172C4×C79—44no—
173C4×C79—42no—
174C4×C79#41312yes—
175C4×C79#54812yes—
176C4×C79D4×Q812yes—
177#22517—791nomaximal, Sylow
178C2×C79—21no—
179C2×C79—21no—
180C2×C79Q8×C22⋊C411yescharacteristic
181C2×C79C2×#41311yescharacteristic
182C2×C79#288911yescharacteristic
183C2×#557D7911yescharacteristic
184Q8×M16—792no—
185#1998—794no—
186#1997—792no—
187#1801—792no—
188#1801—791no—
189C4×#101—792no—
190C4×#101—791no—
191C79#2251711yescharacteristic, complemented, Sylow
192#557—22no—
193#557—24no—
194#557—22no—
195C2×C4⋊Q8Dic7912yes—
196C2×C4⋊Q8C2×D7911yescharacteristic
197C2×#125Dic7914yes—
198C2×#125C2×D7912yes—
199C2×C4×Q8Dic7912yes—
200C2×C4×Q8C2×D7912yes—
201C2×C4×Q8C2×D7911yescharacteristic
202C2×C4×Q8C2×D7911yescharacteristic
203#446—1584no—
204#446—1582no—
205C8×Q8—1582no—
206#423—792no—
207#423—794no—
208C2×#101—791no—
209C2×#101—791no—
210C2×#101—792no—
211C4×M16—792no—
212C4×M16—794no—
213#343—794no—
214#343—792no—
215#343—794no—
216#343—792no—
217C22×Q8C2×Dic7912yes—
218C22×Q8C2×Dic7911yescharacteristic
219C22×Q8C22×D7911yescharacteristic
220C2×M16—792no—
221C2×M16—792no—
222C2×M16—794no—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227#125—28no—
228#125—28no—
229#125—24no—
230#125—24no—
231C4×Q8—24no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8C22⋊D7912yes—
239C4×Q8C22⋊D7914yes—
240C4×Q8C22⋊D7912yes—
241C2×C4⋊C4C2×Dic7914yes—
242C2×C4⋊C4C2×Dic7912yes—
243C2×C4⋊C4C2×Dic7914yes—
244C2×C4⋊C4C2×Dic7912yes—
245C2×C4⋊C4C2×Dic7914yes—
246C2×C4⋊C4C2×Dic7911yescharacteristic
247C2×C4⋊C4C2×Dic7912yes—
248C2×C4⋊C4C22×D7911yescharacteristic
249C2×C4⋊C4C22×D7912yes—
250C2×C42C2×Dic7914yes—
251C2×C42C2×Dic7912yes—
252C2×C42C22×D7912yes—
253C2×C42C22×D7911yescharacteristic
254C4⋊C8—1584no—
255C4⋊C8—1582no—
256C4⋊C8—1584no—
257C4⋊C8—1582no—
258#101—794no—
259#101—1582no—
260#101—1584no—
261#97—1584no—
262#97—1582no—
263C4×C8—1584no—
264C4×C8—1582no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—44no—
271C2×Q8C22⋊Dic7912yes—
272C2×Q8C22⋊Dic7912yes—
273C2×Q8Q8×D7912yes—
274C2×Q8#1192962512yes—
275C22×C4C22×Dic7911yescharacteristic
276C22×C4C22×Dic7912yes—
277C22×C4C22×Dic7912yes—
278C22×C4C22×Dic7914yes—
279C22×C4C22×Dic7912yes—
280C22×C4C22×Dic7911yescharacteristic
281C22×C4C22×Dic7912yes—
282C22×C4C23×D7911yescharacteristic
283M16—794no—
284M16—1582no—
285M16—1584no—
286M16—1582no—
287M16—1584no—
288M16—1582no—
289C2×C8—1584no—
290C2×C8—1582no—
291C2×C8—1582no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4C22⋊Dic7912yes—
301C4⋊C4C22⋊Dic7911yescharacteristic
302C4⋊C4C22⋊Dic7911yescharacteristic
303C4⋊C4C22⋊Dic7914yes—
304C4⋊C4C22⋊Dic7914yes—
305C42—24no—
306C42—24no—
307C42—22no—
308C42—22no—
309C42C2×C22⋊D7914yes—
310C42C2×C22⋊D7912yes—
311C42C2×C22⋊D7914yes—
312C42C2×C22⋊D7912yes—
313C23C23×Dic7911yescharacteristic
314Q8—24no—
315Q8—24no—
316Q8—24no—
317Q8—42no—
318Q8—44no—
319Q8—44no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4C2×C22⋊Dic7912yes—
331C2×C4C2×C22⋊Dic7912yes—
332C2×C4C2×C22⋊Dic7911yescharacteristic
333C2×C4C2×C22⋊Dic7912yes—
334C2×C4C2×C22⋊Dic7914yes—
335C2×C4C2×C22⋊Dic7911yescharacteristic
336C2×C4Q8×Dic7914yes—
337C2×C4C22×C22⋊D7912yes—
338C2×C4C2×Q8×D7911yescharacteristic
339C2×C4C2×#1192962511yescharacteristic
340C2×C4#1457920912yes—
341C8—3164no—
342C8—3162no—
343C8—1584no—
344C22—21no—
345C22—21no—
346C22C22×C22⋊Dic7911yescharacteristic
347C22C2×Q8×Dic7911yescharacteristic
348C22#1909904111yescharacteristic
349C4—24no—
350C4—21no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—44no—
356C4—42no—
357C4#1909854712yes—
358C4#1909922412yes—
359C4Q8×C22⋊D7912yes—
360C2—21no—
361C2—21no—
362C2Q8×C22⋊Dic7911yescharacteristic
363C2C2×#1909854711yescharacteristic
364C2#2377188311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic79
commutator, prime radicalC23×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#557×C79C2
socleC22×C79C22×C22⋊C4
cosocleC22×C4×C79C24
hypercenterC2×#557D79
hypocenterC79#22517
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
43. #14579080Q8semidirect
191. C79#22517semidirect

Maximal subgroups

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

subgroupindex
Q8×#119296082
#237643942
#237643902
#237633962
C4×#145790802
C2×#557×C792
#2251779