Finite Group Databasebeta

← #34033276

Subgroups — #34033276

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#10596842C211yesmaximal, characteristic
2#22627065C211yesmaximal, 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—21no—
15C2×C4×#10596842C2211yescharacteristic
16C2×C4×#10596842C2211yescharacteristic
17(C2×#10596842)⋊C4—22no—
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—22no—
39C4×#10596842—21no—
40C4×#10596842—21no—
41C4×#10596842—21no—
42C4×#10596842C2×C412yes—
43C4×#10596842C2×C412yes—
44C4×#10596842D411yescharacteristic
45C4×#10596842D411yescharacteristic
46D73⋊M16C2×C411yescharacteristic
47Q8×#64408C2×C411yescharacteristic
48Q8×#64408C2×C411yescharacteristic
49D4×#64408—21no—
50D4×#64408—21no—
51D4×#64408C2×C411yescharacteristic
52D4×#64408C2×C411yescharacteristic
53#14556670—21no—
54#14556670C2×C411yescharacteristic
55C2×C4×#64408—21no—
56C2×C4×#64408—21no—
57C2×C4×#64408—21no—
58C2×C4×#64408C2×C411yescharacteristic
59C2×C4×#64408C2311yescharacteristic
60#14556656C812yescomplemented
61#14556655C812yescomplemented
62#14556645—21no—
63C8×#64408—21no—
64#41×D73C2×C411yescharacteristic
65M16×D73C2×C411yescharacteristic
66C42×D73C2×C411yescharacteristic
67#14556519C811yescharacteristic, complemented
68#14556518C811yescharacteristic, complemented
69#14556517C811yescharacteristic, complemented
70#104×C73C811yescharacteristic, complemented
71C2×#10596842—22no—
72C2×#10596842—22no—
73C2×#10596842—22no—
74C2×#10596842—42no—
75C2×#10596842—44no—
76C2×#10596842—42no—
77C2×#10596842C22⋊C412yes—
78C2×#10596842C22⋊C412yes—
79C22×#64408—21no—
80C22×#64408—21no—
81C22×#64408—21no—
82C22×#64408C2×D411yescharacteristic
83C22×#64408#4111yescharacteristic
84#11924038C2×C812yescomplemented
85#11924037—21no—
86#11924036—21no—
87C4⋊#64408—21no—
88C4⋊#64408—21no—
89C4⋊#64408C4211yescharacteristic
90C4⋊#64408C4211yescharacteristic
91C22⋊#64408—21no—
92C22⋊#64408—21no—
93C22⋊#64408—21no—
94C4×#64408—21no—
95C4×#64408—21no—
96C4×#64408—21no—
97C4×#64408—21no—
98C4×#64408—42no—
99C4×#64408—42no—
100C4×#64408C2×C812yescomplemented
101C4×#64408C22×C411yescharacteristic
102C4×#64408C22×C411yescharacteristic
103C4×#64408C2×D411yescharacteristic
104C4×#64408#4111yescharacteristic
105#11924029C2×C811yescharacteristic
106#11924028—21no—
107#11924028C2×C811yescharacteristic
108#11924027—21no—
109#11924027C2×C811yescharacteristic
110Q8×D73C4211yescharacteristic
111D4×D73—21no—
112D4×D73C4211yescharacteristic
113C2×C4×D73—21no—
114C2×C4×D73—21no—
115C2×C4×D73C22×C411yescharacteristic
116#11924012C2×C811yescharacteristic, complemented
117#11924011—21no—
118C8×D73—21no—
119C4×Dic73—21no—
120C4×Dic73C2×C811yescharacteristic, complemented
121#41×C73C2×C811yescharacteristic
122M16×C73C2×C811yescharacteristic, complemented
123C42×C73C2×C811yescharacteristic, complemented
124C22×D73—21no—
125C22×D73C4×D411yescharacteristic
126C2×#64408—22no—
127C2×#64408—22no—
128C2×#64408—21no—
129C2×#64408—21no—
130C2×#64408—22no—
131C2×#64408—41no—
132C2×#64408—41no—
133C2×#64408—42no—
134C2×#64408—42no—
135C2×#64408C2×C22⋊C411yescharacteristic
136C2×#64408#11711yescharacteristic
137C2×#64408C4×D411yescharacteristic
138C2×#64408C4×D411yescharacteristic
139#10596842—22no—
140#10596842—22no—
141#10596842—42no—
142#10596842—42no—
143#10596842—42no—
144Q8×C73C4×C811yescharacteristic
145D4×C73—21no—
146D4×C73C4×C811yescharacteristic
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73C22×C811yescharacteristic
150C22⋊D73—21no—
151C22⋊D73—21no—
152C22⋊D73—21no—
153C2×Dic73—21no—
154C2×Dic73—21no—
155C2×Dic73#13111yescharacteristic
156D292—21no—
157D292C4×C811yescharacteristic
158C4×D73—21no—
159C4×D73—21no—
160C4×D73—21no—
161C4×D73—41no—
162C4×D73—41no—
163C4×D73C2×C4211yescharacteristic
164C4×D73C4×D411yescharacteristic
165C4×D73C22×C811yescharacteristic
166C4×D73#13111yescharacteristic
167Dic146—21no—
168Dic146C4×C811yescharacteristic
169#10596833—22no—
170C8×C73—21no—
171#10596831—21no—
172C22×C73—21no—
173C22×C73C8×D411yescharacteristic
174C2×D73—21no—
175C2×D73—21no—
176C2×D73—21no—
177C2×D73—41no—
178C2×D73—41no—
179C2×D73C4×C22⋊C411yescharacteristic
180C2×D73C8×D411yescharacteristic
181#64408—21no—
182#64408—21no—
183#64408—42no—
184#64408—41no—
185#64408—41no—
186#64408#42112yes—
187#64408#44412yes—
188C4×C73—21no—
189C4×C73—41no—
190C4×C73C2×C4×C811yescharacteristic
191C4×C73C8×D411yescharacteristic
192Dic73—21no—
193Dic73—41no—
194Dic73#40511yescharacteristic
195Dic73C8×D411yescharacteristic
196#8191—731nomaximal, Sylow
197C2×C73—21no—
198C2×C73—41no—
199C2×C73C8×C22⋊C411yescharacteristic
200D73—21no—
201D73—41no—
202D73#175911yescharacteristic
203D73#176511yescharacteristic
204C8×#41—731no—
205C4×#104—731no—
206#1744—731no—
207#1476—731no—
208C8×M16—731no—
209#1278—731no—
210#1276—731no—
211C73#819111yescharacteristic, complemented, Sylow
212C4×#41—731no—
213C8×Q8—732no—
214C8×D4—732no—
215C8×D4—1462no—
216#431—731no—
217#431—731no—
218#431—1461no—
219C2×#104—731no—
220C2×#104—731no—
221C4×M16—731no—
222C2×C4×C8—731no—
223C2×C4×C8—731no—
224C2×C4×C8—1461no—
225C43—731no—
226#339—731no—
227#339—731no—
228#336—1462no—
229Q8⋊C8—732no—
230D4⋊C8—732no—
231#322—1461no—
232C82—1461no—
233C2×#41—731no—
234C2×M16—731no—
235C2×M16—731no—
236C22×C8—732no—
237C22×C8—1462no—
238C22×C8—1461no—
239C4×Q8—731no—
240C4×Q8—731no—
241C4×D4—731no—
242C4×D4—731no—
243C4×D4—1461no—
244C4×D4—1461no—
245#117—731no—
246#117—1461no—
247C2×C42—731no—
248C2×C42—731no—
249C2×C42—731no—
250C2×C42—1461no—
251C2×C42—1461no—
252C2×C42—1461no—
253C4⋊C8—732no—
254C4⋊C8—732no—
255C4⋊C8—1462no—
256#104—732no—
257#104—732no—
258#104—731no—
259#104—731no—
260#104—731no—
261#104#1059684211yescharacteristic, complemented
262C22⋊C8—1462no—
263C22⋊C8—1461no—
264C22⋊C8—1461no—
265#97—1461no—
266C4×C8—732no—
267C4×C8—731no—
268C4×C8—731no—
269C4×C8—732no—
270C4×C8—1461no—
271C4×C8—1461no—
272C4×C8—1461no—
273C4×C8—1461no—
274C4×C8—1462no—
275#41—731no—
276#41—731no—
277#41—731no—
278#41—1461no—
279#41—1461no—
280#41C2×#1059684211yescharacteristic
281C2×Q8—731no—
282C2×D4—731no—
283C2×D4—1461no—
284C22×C4—731no—
285C22×C4—731no—
286C22×C4—731no—
287C22×C4—1461no—
288C22×C4—1461no—
289C22×C4—1461no—
290C22×C4—1461no—
291C22×C4—1461no—
292M16—732no—
293M16—731no—
294M16—1461no—
295M16—1461no—
296M16C2×#1059684211yescharacteristic, complemented
297C2×C8—2922no—
298C2×C8—2924no—
299C2×C8—2922no—
300C2×C8—732no—
301C2×C8—732no—
302C2×C8—1461no—
303C2×C8—1462no—
304C2×C8—1462no—
305C2×C8—1462no—
306C2×C8—1461no—
307C4⋊C4—731no—
308C4⋊C4—731no—
309C4⋊C4—1461no—
310C4⋊C4—1461no—
311C22⋊C4—1461no—
312C22⋊C4—1461no—
313C22⋊C4—1461no—
314C42—2922no—
315C42—2922no—
316C42—731no—
317C42—731no—
318C42—731no—
319C42—731no—
320C42—732no—
321C42—731no—
322C42—1461no—
323C42—1461no—
324C42—1461no—
325C42—1461no—
326C42—1461no—
327C42C2×#1059684211yescharacteristic, complemented
328C23—731no—
329C23—1461no—
330Q8—731no—
331Q8—1461no—
332Q8C4×#1059684211yescharacteristic
333D4—21no—
334D4—731no—
335D4—1461no—
336D4—1461no—
337D4—1461no—
338D4—1461no—
339D4C4×#1059684211yescharacteristic
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2921no—
343C2×C4—2921no—
344C2×C4—2921no—
345C2×C4—2921no—
346C2×C4—2922no—
347C2×C4—2922no—
348C2×C4—731no—
349C2×C4—731no—
350C2×C4—731no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—731no—
354C2×C4—731no—
355C2×C4—731no—
356C2×C4—731no—
357C2×C4—1461no—
358C2×C4—1461no—
359C2×C4—1462no—
360C2×C4—1461no—
361C2×C4—1461no—
362C2×C4—1461no—
363C2×C4—1461no—
364C2×C4—1462no—
365C2×C4—1462no—
366C2×C4—1461no—
367C2×C4C22×#1059684211yescharacteristic
368C8—21no—
369C8—2922no—
370C8—2922no—
371C8—2922no—
372C8—1462no—
373C8—1462no—
374C8—1462no—
375C8—1461no—
376C22—21no—
377C22—2921no—
378C22—2921no—
379C22—731no—
380C22—731no—
381C22—1461no—
382C22—1461no—
383C22—1461no—
384C22D4×#1059684211yescharacteristic
385C4—21no—
386C4—41no—
387C4—2921no—
388C4—2921no—
389C4—2921no—
390C4—2922no—
391C4—732no—
392C4—732no—
393C4—731no—
394C4—731no—
395C4—1461no—
396C4—1461no—
397C4—1461no—
398C4C2×C4×#1059684211yescharacteristic
399C4D4×#1059684211yescharacteristic
400C2—21no—
401C2—41no—
402C2—2921no—
403C2—731no—
404C2—731no—
405C2—1461no—
406C2C22⋊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
hypocenterC73#8191
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. #22627065C2semidirect
4. M16×#10596842C2semidirect
5. #22626938C2semidirect
6. #22626936C2semidirect
8. Q8×#10596842C4semidirect
10. D4×#10596842C4semidirect
18. Q8⋊#10596842C4semidirect
19. D4⋊#10596842C4semidirect
60. #14556656C8semidirect
61. #14556655C8semidirect
67. #14556519C8semidirect
68. #14556518C8semidirect
69. #14556517C8semidirect
70. #104×C73C8semidirect
84. #11924038C2×C8semidirect
100. C4×#64408C2×C8semidirect
116. #11924012C2×C8semidirect
120. C4×Dic73C2×C8semidirect
122. M16×C73C2×C8semidirect
123. C42×C73C2×C8semidirect
211. C73#8191semidirect
261. #104#10596842semidirect
296. M16C2×#10596842semidirect
327. C42C2×#10596842semidirect

Maximal subgroups

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

subgroupindex
#41×#105968422
#226270652
#226270492
M16×#105968422
#226269382
#226269362
#104×#644082
#819173