Finite Group Databasebeta

← D97⋊#413 #27158344

Subgroups — D97⋊#413 #27158344

Order 12416.

756 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D97⋊M16C212yesmaximal, complemented
2#20261722C214yesmaximal, complemented
3#20261721C214yesmaximal, complemented
4D97⋊#101C212yesmaximal, complemented
5D97⋊#100C212yesmaximal, complemented
6C2×#41×D97C211yesmaximal, characteristic
7C2×#13095521—22no—
8D97⋊M16—22no—
9D97⋊M16—22no—
10D97⋊M16C2214yes—
11D97⋊M16C2214yescomplemented
12C2×#13095520C2214yescomplemented
13C2×#13095519—22no—
14#15690871C2218yescomplemented
15#15690870C2218yescomplemented
16C2×#13095512C411yescharacteristic
17C2×#13095511C411yescharacteristic
18C2×#13095511C2212yes—
19C2×#13095510C411yescharacteristic
20C2×#13095510C2212yes—
21#41×D97—22no—
22#41×D97—22no—
23#41×D97—22no—
24#41×D97—22no—
25C2×Q8×D97C2211yescharacteristic
26C2×D4×D97C412yes—
27C2×D4×D97C2211yescharacteristic
28C22×C4×D97C412yes—
29C22×C4×D97C2211yescharacteristic
30C2×#41×C97C411yescharacteristic
31#13095521—24no—
32#13095521—24no—
33#13095520—24no—
34#13095520C2318yes—
35#13095519—24no—
36#13095519—24no—
37C2×#11715897—24no—
38C23×D97C2×C412yes—
39C23×D97C2×C411yescharacteristic
40#13095512—22no—
41#13095512—22no—
42#13095511—22no—
43#13095511—22no—
44#13095511—24no—
45#13095511—24no—
46#13095510—24no—
47#13095510—24no—
48#13095510—22no—
49#13095510—22no—
50C2×Dic194C2×C411yescharacteristic
51C2×Dic194C2312yes—
52Q8×D97—22no—
53Q8×D97—22no—
54C2×D388C2×C412yes—
55C2×D388C2×C411yescharacteristic
56C2×C22⋊D97C2×C412yes—
57C2×C22⋊D97C2×C412yes—
58C2×C22⋊D97C2312yes—
59D4×D97—24no—
60D4×D97—24no—
61D4×D97—22no—
62D4×D97—22no—
63C22×Dic97C2×C412yes—
64C22×Dic97C2311yescharacteristic
65C2×C4×D97—22no—
66C2×C4×D97—22no—
67C2×C4×D97—22no—
68C2×C4×D97—22no—
69C2×C4×D97C2×C412yes—
70C2×C4×D97C2×C412yes—
71C2×C4×D97C2×C412yes—
72C2×C4×D97C2×C411yescharacteristic
73C2×C4×D97C2×C411yescharacteristic
74C2×C4×D97D412yes—
75C2×C4×D97D412yes—
76C2×C4×D97D412yes—
77C2×C4×D97D412yes—
78C2×C4×D97C2311yescharacteristic
79C2×C4×D97C2311yescharacteristic
80#41×C97—22no—
81#41×C97—22no—
82C2×Q8×C97C2×C411yescharacteristic
83C2×D4×C97C2×C412yes—
84C2×D4×C97C2×C411yescharacteristic
85C22×C4×C97C2×C412yes—
86C22×C4×C97C2×C411yescharacteristic
87C23×C97C22×C412yes—
88C23×C97C22×C411yescharacteristic
89C22×D97—22no—
90C22×D97—22no—
91C22×D97—42no—
92C22×D97—44no—
93C22×D97C22⋊C412yes—
94C22×D97C22⋊C412yes—
95C22×D97C22×C411yescharacteristic
96C22×D97C22×C412yes—
97C22×D97C22×C411yescharacteristic
98Q8×C97—22no—
99D4×C97—24no—
100D4×C97—22no—
101C2×C4×C97—22no—
102C2×C4×C97—22no—
103C2×C4×C97C22⋊C412yes—
104C2×C4×C97C22⋊C412yes—
105C2×C4×C97C22×C412yes—
106C2×C4×C97C22×C411yescharacteristic
107C2×C4×C97C22×C411yescharacteristic
108C22⋊D97—24no—
109C22⋊D97—24no—
110C22⋊D97—24no—
111C2×Dic97—22no—
112C2×Dic97—22no—
113C2×Dic97C22×C412yes—
114C2×Dic97C22×C411yescharacteristic
115C2×Dic97C2×D412yes—
116C2×Dic97C2×D412yes—
117C2×Dic97C2411yescharacteristic
118D388—24no—
119D388—22no—
120C4×D97—22no—
121C4×D97—22no—
122C4×D97—22no—
123C4×D97—22no—
124C4×D97—22no—
125C4×D97—22no—
126C4×D97C22⋊C411yescharacteristic
127C4×D97C22⋊C411yescharacteristic
128C4×D97C22⋊C411yescharacteristic
129C4×D97C22⋊C411yescharacteristic
130C4×D97C22⋊C412yes—
131C4×D97C22⋊C412yes—
132C4×D97C2×D412yes—
133C4×D97C2×D412yes—
134C4×D97C2×D412yes—
135C4×D97C2×D412yes—
136Dic194—22no—
137Dic194—24no—
138#11715897—28no—
139C22×C97—22no—
140C22×C97—41no—
141C22×C97—42no—
142C22×C97C2×C22⋊C412yes—
143C22×C97C23×C411yescharacteristic
144C2×D97—22no—
145C2×D97—24no—
146C2×D97—22no—
147C2×D97—42no—
148C2×D97—41no—
149C2×D97—44no—
150C2×D97—44no—
151C2×D97—42no—
152C2×D97C2×C22⋊C411yescharacteristic
153C2×D97C2×C22⋊C412yes—
154C2×D97C2×C22⋊C411yescharacteristic
155C4×C97—22no—
156C4×C97C2×C22⋊C412yes—
157C4×C97C2×C22⋊C411yescharacteristic
158C4×C97C2×C22⋊C411yescharacteristic
159Dic97—22no—
160Dic97C2×C22⋊C411yescharacteristic
161Dic97C2×C22⋊C411yescharacteristic
162Dic97C22×D412yes—
163C2×C97—21no—
164C2×C97—22no—
165C2×C97—42no—
166C2×C97C22×C22⋊C411yescharacteristic
167D97—21no—
168D97—22no—
169D97—42no—
170D97#41312yescomplemented
171C2×#413—971nomaximal, Sylow
172C97C2×#41311yescharacteristic, complemented, Sylow
173C22×#41—971no—
174C22×M16—972no—
175#413—974no—
176#413—974no—
177C2×#101—972no—
178C2×#100—972no—
179C2×#41—971no—
180C2×#41—971no—
181C2×#41—972no—
182C2×#41—972no—
183C2×#41—971no—
184C2×#41—1942no—
185C2×#41—1942no—
186C2×#41—1942no—
187C2×#41—1942no—
188C2×#41#8760411yescharacteristic
189C22×Q8—971no—
190C22×D4—972no—
191C22×D4—971no—
192C23×C4—971no—
193C23×C4—972no—
194C2×M16—974no—
195C2×M16—974no—
196C2×M16—974no—
197C2×M16—1942no—
198C2×M16—1942no—
199C2×M16—1942no—
200C22×C8—1942no—
201#101—978no—
202#100—978no—
203C24—971no—
204C24—972no—
205#41—22no—
206#41—22no—
207#41—1944no—
208#41—1944no—
209#41—1944no—
210#41—1944no—
211#41—1942no—
212#41—1942no—
213#41—1942no—
214#41—1942no—
215#41—1942no—
216#41—1942no—
217C2×Q8—971no—
218C2×Q8—972no—
219C2×Q8—1942no—
220C2×Q8—1942no—
221C2×Q8C2×#8760411yescharacteristic
222C2×D4—972no—
223C2×D4—972no—
224C2×D4—972no—
225C2×D4—972no—
226C2×D4—971no—
227C2×D4—1942no—
228C2×D4—1942no—
229C2×D4—1944no—
230C2×D4—1944no—
231C2×D4C2×#8760412yes—
232C2×D4C2×#8760411yescharacteristic
233C22×C4—971no—
234C22×C4—972no—
235C22×C4—972no—
236C22×C4—972no—
237C22×C4—972no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—971no—
241C22×C4—971no—
242C22×C4—972no—
243C22×C4—972no—
244C22×C4—972no—
245C22×C4—972no—
246C22×C4—1942no—
247C22×C4—1942no—
248C22×C4—1942no—
249C22×C4—1942no—
250C22×C4C2×#8760412yes—
251C22×C4C2×#8760411yescharacteristic
252M16—978no—
253M16—1944no—
254M16—1944no—
255M16—1944no—
256C2×C8—1944no—
257C2×C8—1944no—
258C2×C8—1944no—
259C23—972no—
260C23—971no—
261C23—972no—
262C23—971no—
263C23—972no—
264C23—3884no—
265C23—3882no—
266C23—1942no—
267C23—1942no—
268C23C22×#8760412yes—
269C23C22×#8760411yescharacteristic
270Q8—22no—
271Q8—1944no—
272Q8—1942no—
273D4—24no—
274D4—22no—
275D4—1944no—
276D4—1944no—
277D4—1944no—
278D4—1944no—
279D4—1942no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—972no—
283C2×C4—972no—
284C2×C4—971no—
285C2×C4—971no—
286C2×C4—972no—
287C2×C4—972no—
288C2×C4—972no—
289C2×C4—971no—
290C2×C4—971no—
291C2×C4—972no—
292C2×C4—972no—
293C2×C4—972no—
294C2×C4—971no—
295C2×C4—971no—
296C2×C4—972no—
297C2×C4—1942no—
298C2×C4—1942no—
299C2×C4—1942no—
300C2×C4—1942no—
301C2×C4—1942no—
302C2×C4—1942no—
303C2×C4—1942no—
304C2×C4—1942no—
305C2×C4C22⋊#8760412yes—
306C2×C4C22⋊#8760412yes—
307C2×C4C22×#8760412yes—
308C2×C4C22×#8760411yescharacteristic
309C2×C4C22×#8760411yescharacteristic
310C8—1948no—
311C22—22no—
312C22—41no—
313C22—42no—
314C22—971no—
315C22—971no—
316C22—972no—
317C22—3882no—
318C22—3881no—
319C22—3882no—
320C22—3884no—
321C22—3884no—
322C22—1942no—
323C22—1944no—
324C22—1942no—
325C22C2×C22⋊#8760412yes—
326C22C23×#8760411yescharacteristic
327C4—22no—
328C4—972no—
329C4—971no—
330C4—971no—
331C4—1942no—
332C4C2×C22⋊#8760411yescharacteristic
333C4C2×C22⋊#8760412yes—
334C4C2×C22⋊#8760411yescharacteristic
335C2—21no—
336C2—22no—
337C2—42no—
338C2—972no—
339C2—3882no—
340C2—1941no—
341C2—1942no—
342C2C22×C22⋊#8760411yescharacteristic
subgroupG/H
centerC4C2×C22⋊#87604
commutatorC22×C97C23×C4
FrattiniC22C23×#87604
FittingC2×#41×C97C4
socleC2×C97C22×C22⋊C4
prime radicalC2×#41×D97C2
cosocleC2×Dic97C24
hypercenterC2×#41#87604
hypocenterC97C2×#413
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D97⋊M16C2semidirect
2. #20261722C2semidirect
3. #20261721C2semidirect
4. D97⋊#101C2semidirect
5. D97⋊#100C2semidirect
11. D97⋊M16C22semidirect
12. C2×#13095520C22semidirect
14. #15690871C22semidirect
15. #15690870C22semidirect
170. D97#413semidirect
172. C97C2×#413semidirect

Maximal subgroups

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

subgroupindex
C2×D97⋊M162
#202617222
#202617212
D97⋊#1012
D97⋊#1002
C2×#41×D972
C2×#41397