Finite Group Databasebeta

← #104×#10596842 #34033275

Subgroups — #104×#10596842 #34033275

Order 18688.

554 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#10596842C211yesmaximal, characteristic
2C42×#10596842C211yesmaximal, characteristic, complemented
3#22627049C211yesmaximal, characteristic
4M16×#10596842C211yesmaximal, characteristic, complemented
5#22626938C211yesmaximal, characteristic, complemented
6#22626936C211yesmaximal, characteristic, complemented
7#104×#64408C211yesmaximal, characteristic
8Q8×#10596842C412yescomplemented
9D4×#10596842—22no—
10D4×#10596842C412yescomplemented
11#17989700—21no—
12#17989700C2211yescharacteristic
13#17989700C2211yescharacteristic
14C2×C4×#10596842—22no—
15C2×C4×#10596842—21no—
16C2×C4×#10596842C2211yescharacteristic
17C2×C4×#10596842C2211yescharacteristic
18Q8⋊#10596842C412yescomplemented
19D4⋊#10596842C412yescomplemented
20#17989671—21no—
21C8×#10596842—21no—
22#41×#64408C2211yescharacteristic
23D73⋊#104C411yescharacteristic
24M16×#64408C2211yescharacteristic
25C42×#64408C2211yescharacteristic
26#17989459C411yescharacteristic
27#17989458C411yescharacteristic
28#104×D73C411yescharacteristic
29C22×#10596842—22no—
30C22×#10596842—21no—
31C22×#10596842D412yes—
32C4⋊#10596842—22no—
33C4⋊#10596842C2×C412yes—
34C4⋊#10596842C2×C412yes—
35C22⋊#10596842—22no—
36C22⋊#10596842—21no—
37C22⋊#10596842—21no—
38C4×#10596842—28no—
39C4×#10596842—22no—
40C4×#10596842—21no—
41C4×#10596842—21no—
42C4×#10596842—21no—
43C4×#10596842C2×C412yes—
44C4×#10596842C2×C412yes—
45C4×#10596842D411yescharacteristic
46C4×#10596842D411yescharacteristic
47D73⋊M16C2×C411yescharacteristic
48Q8×#64408C2×C411yescharacteristic
49Q8×#64408C2×C411yescharacteristic
50D4×#64408—21no—
51D4×#64408—21no—
52D4×#64408C2×C411yescharacteristic
53D4×#64408C2×C411yescharacteristic
54#14556670—21no—
55#14556670C2×C411yescharacteristic
56C2×C4×#64408—21no—
57C2×C4×#64408—21no—
58C2×C4×#64408—21no—
59C2×C4×#64408C2×C411yescharacteristic
60C2×C4×#64408C2311yescharacteristic
61#14556656C812yescomplemented
62#14556655C812yescomplemented
63#14556645—21no—
64C8×#64408—21no—
65#41×D73C2×C411yescharacteristic
66M16×D73C2×C411yescharacteristic
67C42×D73C2×C411yescharacteristic
68#14556519C811yescharacteristic, complemented
69#14556518C811yescharacteristic, complemented
70#14556517C811yescharacteristic, complemented
71#104×C73C811yescharacteristic, complemented
72C2×#10596842—24no—
73C2×#10596842—22no—
74C2×#10596842—24no—
75C2×#10596842—22no—
76C2×#10596842—22no—
77C2×#10596842—44no—
78C2×#10596842C22⋊C412yes—
79C2×#10596842C22⋊C412yes—
80C22×#64408—21no—
81C22×#64408—21no—
82C22×#64408—21no—
83C22×#64408C2×D411yescharacteristic
84C22×#64408#4111yescharacteristic
85#11924038C2×C812yescomplemented
86#11924037—21no—
87#11924036—21no—
88C4⋊#64408—21no—
89C4⋊#64408—21no—
90C4⋊#64408C4211yescharacteristic
91C4⋊#64408C4211yescharacteristic
92C22⋊#64408—21no—
93C22⋊#64408—21no—
94C22⋊#64408—21no—
95C4×#64408—24no—
96C4×#64408—24no—
97C4×#64408—21no—
98C4×#64408—21no—
99C4×#64408—21no—
100C4×#64408—21no—
101C4×#64408C2×C812yescomplemented
102C4×#64408C22×C411yescharacteristic
103C4×#64408C22×C411yescharacteristic
104C4×#64408C2×D411yescharacteristic
105C4×#64408#4111yescharacteristic
106#11924029C2×C811yescharacteristic
107#11924028—21no—
108#11924028C2×C811yescharacteristic
109#11924027—21no—
110#11924027C2×C811yescharacteristic
111Q8×D73C4211yescharacteristic
112D4×D73—21no—
113D4×D73C4211yescharacteristic
114C2×C4×D73—21no—
115C2×C4×D73—21no—
116C2×C4×D73C22×C411yescharacteristic
117#11924012C2×C811yescharacteristic, complemented
118#11924011—21no—
119C8×D73—21no—
120C4×Dic73—21no—
121C4×Dic73C2×C811yescharacteristic, complemented
122#41×C73C2×C811yescharacteristic
123M16×C73C2×C811yescharacteristic, complemented
124C42×C73C2×C811yescharacteristic, complemented
125C22×D73—21no—
126C22×D73C4×D411yescharacteristic
127C2×#64408—22no—
128C2×#64408—22no—
129C2×#64408—22no—
130C2×#64408—22no—
131C2×#64408—22no—
132C2×#64408—21no—
133C2×#64408—21no—
134C2×#64408—42no—
135C2×#64408—42no—
136C2×#64408C2×C22⋊C411yescharacteristic
137C2×#64408#11711yescharacteristic
138C2×#64408C4×D411yescharacteristic
139C2×#64408C4×D411yescharacteristic
140#10596842—24no—
141#10596842—22no—
142#10596842—42no—
143#10596842—42no—
144#10596842#10414yesdirect product factor
145Q8×C73C4×C811yescharacteristic, complemented
146D4×C73—21no—
147D4×C73C4×C811yescharacteristic, complemented
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73C22×C811yescharacteristic
151C22⋊D73—21no—
152C22⋊D73—21no—
153C22⋊D73—21no—
154C2×Dic73—21no—
155C2×Dic73—21no—
156C2×Dic73#13111yescharacteristic
157D292—21no—
158D292C4×C811yescharacteristic, complemented
159C4×D73—22no—
160C4×D73—22no—
161C4×D73—21no—
162C4×D73—21no—
163C4×D73—21no—
164C4×D73C2×C4211yescharacteristic
165C4×D73C4×D411yescharacteristic
166C4×D73C22×C811yescharacteristic
167C4×D73#13111yescharacteristic
168Dic146—21no—
169Dic146C4×C811yescharacteristic, complemented
170#10596833—22no—
171C8×C73—21no—
172#10596831—21no—
173C22×C73—21no—
174C22×C73C8×D411yescharacteristic
175C2×D73—21no—
176C2×D73—21no—
177C2×D73—21no—
178C2×D73—41no—
179C2×D73—41no—
180C2×D73C4×C22⋊C411yescharacteristic
181C2×D73C8×D411yescharacteristic
182#64408—24no—
183#64408—21no—
184#64408—21no—
185#64408—41no—
186#64408—41no—
187#64408C2×#10412yes—
188#64408#44312yes—
189C4×C73—22no—
190C4×C73—21no—
191C4×C73C2×C4×C811yescharacteristic
192C4×C73C8×D411yescharacteristic
193Dic73—22no—
194Dic73—21no—
195Dic73#40511yescharacteristic
196Dic73C8×D411yescharacteristic
197C8×#104—731nomaximal, Sylow
198C2×C73—21no—
199C2×C73—41no—
200C2×C73C8×C22⋊C411yescharacteristic
201D73—21no—
202D73—41no—
203D73C4×#10411yescharacteristic
204D73#176411yescharacteristic
205C8×#41—731no—
206C4×#104—731no—
207C42×C8—731no—
208#1476—731no—
209C8×M16—731no—
210#1278—731no—
211#1276—731no—
212C73C8×#10411yescharacteristic, complemented, Sylow
213C4×#41—731no—
214C8×Q8—732no—
215C8×D4—732no—
216C8×D4—1462no—
217#431—731no—
218#431—731no—
219#431—1461no—
220C2×#104—731no—
221C2×#104—731no—
222C4×M16—731no—
223C2×C4×C8—731no—
224C2×C4×C8—731no—
225C2×C4×C8—1461no—
226C2×C4×C8—1462no—
227C43—731no—
228#339—731no—
229#339—731no—
230Q8⋊C8—732no—
231D4⋊C8—732no—
232#322—1461no—
233C82—1461no—
234C2×#41—731no—
235C2×M16—731no—
236C2×M16—731no—
237C22×C8—732no—
238C22×C8—1462no—
239C22×C8—1461no—
240C4×Q8—731no—
241C4×Q8—731no—
242C4×D4—731no—
243C4×D4—731no—
244C4×D4—1461no—
245C4×D4—1461no—
246#117—731no—
247#117—1461no—
248C2×C42—731no—
249C2×C42—731no—
250C2×C42—731no—
251C2×C42—1461no—
252C2×C42—1461no—
253C2×C42—1461no—
254C4⋊C8—732no—
255C4⋊C8—732no—
256C4⋊C8—1462no—
257#104—732no—
258#104—732no—
259#104—731no—
260#104—731no—
261#104—731no—
262#104#1059684211yescharacteristic, direct product factor
263C22⋊C8—1462no—
264C22⋊C8—1461no—
265C22⋊C8—1461no—
266#97—1461no—
267C4×C8—732no—
268C4×C8—731no—
269C4×C8—731no—
270C4×C8—732no—
271C4×C8—1461no—
272C4×C8—1461no—
273C4×C8—1461no—
274C4×C8—1461no—
275C4×C8—1462no—
276C4×C8—1468no—
277#41—731no—
278#41—731no—
279#41—731no—
280#41—1461no—
281#41—1461no—
282#41C2×#1059684211yescharacteristic
283C2×Q8—731no—
284C2×D4—731no—
285C2×D4—1461no—
286C22×C4—731no—
287C22×C4—731no—
288C22×C4—731no—
289C22×C4—1461no—
290C22×C4—1461no—
291C22×C4—1461no—
292C22×C4—1461no—
293C22×C4—1461no—
294M16—732no—
295M16—731no—
296M16—1461no—
297M16—1461no—
298M16C2×#1059684211yescharacteristic, complemented
299C2×C8—2924no—
300C2×C8—732no—
301C2×C8—732no—
302C2×C8—1461no—
303C2×C8—1462no—
304C2×C8—1462no—
305C2×C8—1464no—
306C2×C8—1461no—
307C2×C8—1462no—
308C2×C8—1464no—
309C4⋊C4—731no—
310C4⋊C4—731no—
311C4⋊C4—1461no—
312C4⋊C4—1461no—
313C22⋊C4—1461no—
314C22⋊C4—1461no—
315C22⋊C4—1461no—
316C42—731no—
317C42—731no—
318C42—731no—
319C42—731no—
320C42—732no—
321C42—731no—
322C42—1461no—
323C42—1461no—
324C42—1461no—
325C42—1461no—
326C42—1461no—
327C42—1464no—
328C42—1464no—
329C42C2×#1059684211yescharacteristic, complemented
330C23—731no—
331C23—1461no—
332Q8—731no—
333Q8—1461no—
334Q8C4×#1059684211yescharacteristic, complemented
335D4—21no—
336D4—731no—
337D4—1461no—
338D4—1461no—
339D4—1461no—
340D4—1461no—
341D4C4×#1059684211yescharacteristic, complemented
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2922no—
345C2×C4—2922no—
346C2×C4—731no—
347C2×C4—731no—
348C2×C4—731no—
349C2×C4—731no—
350C2×C4—731no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—731no—
354C2×C4—731no—
355C2×C4—1461no—
356C2×C4—1461no—
357C2×C4—1462no—
358C2×C4—1462no—
359C2×C4—1461no—
360C2×C4—1461no—
361C2×C4—1461no—
362C2×C4—1462no—
363C2×C4—1462no—
364C2×C4—1462no—
365C2×C4—1462no—
366C2×C4—1462no—
367C2×C4—1461no—
368C2×C4—1461no—
369C2×C4C22×#1059684211yescharacteristic
370C8—21no—
371C8—2922no—
372C8—2922no—
373C8—734no—
374C8—1462no—
375C8—1462no—
376C8—1464no—
377C8—1461no—
378C22—21no—
379C22—2921no—
380C22—2921no—
381C22—731no—
382C22—731no—
383C22—1461no—
384C22—1461no—
385C22—1461no—
386C22D4×#1059684211yescharacteristic
387C4—22no—
388C4—21no—
389C4—2921no—
390C4—2921no—
391C4—732no—
392C4—732no—
393C4—731no—
394C4—731no—
395C4—1462no—
396C4—1461no—
397C4—1461no—
398C4—1464no—
399C4—1461no—
400C4C2×C4×#1059684211yescharacteristic
401C4D4×#1059684211yescharacteristic
402C2—21no—
403C2—41no—
404C2—2921no—
405C2—731no—
406C2—731no—
407C2—1461no—
408C2C22⋊C4×#1059684211yescharacteristic
subgroupG/H
centerC4D4×#10596842
commutatorC4×C73C2×C4×C8
FrattiniC2×C4C22×#10596842
Fitting#104×C73C8
socleC2×C73C8×C22⋊C4
prime radical#41×D73C2×C4
cosocleC2×C4×#64408C23
hypercenter#104#10596842
hypocenterC73C8×#104
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. C42×#10596842C2semidirect
4. M16×#10596842C2semidirect
5. #22626938C2semidirect
6. #22626936C2semidirect
8. Q8×#10596842C4semidirect
10. D4×#10596842C4semidirect
18. Q8⋊#10596842C4semidirect
19. D4⋊#10596842C4semidirect
61. #14556656C8semidirect
62. #14556655C8semidirect
68. #14556519C8semidirect
69. #14556518C8semidirect
70. #14556517C8semidirect
71. #104×C73C8semidirect
85. #11924038C2×C8semidirect
101. C4×#64408C2×C8semidirect
117. #11924012C2×C8semidirect
121. C4×Dic73C2×C8semidirect
123. M16×C73C2×C8semidirect
124. C42×C73C2×C8semidirect
144. #10596842#104direct (N × H)
145. Q8×C73C4×C8semidirect
147. D4×C73C4×C8semidirect
158. D292C4×C8semidirect
169. Dic146C4×C8semidirect
212. C73C8×#104semidirect
262. #104#10596842direct (N × H)
298. M16C2×#10596842semidirect
329. C42C2×#10596842semidirect
334. Q8C4×#10596842semidirect
341. D4C4×#10596842semidirect

Maximal subgroups

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

subgroupindex
#41×#105968422
C42×#105968422
#226270492
M16×#105968422
#226269382
#226269362
#104×#644082
C8×#10473