Finite Group Databasebeta

← #34002133

Subgroups — #34002133

Order 18688.

700 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626607C211yesmaximal, characteristic
2C2×C4×#11924037C211yesmaximal, characteristic
3#22624764C214yesmaximal, complemented
4C2×M16×Dic73C211yesmaximal, characteristic
5C22×#11924037—21no—
6#17989608C412yes—
7#17989607—21no—
8#17989604—21no—
9#17989603C2212yes—
10#17989590C412yes—
11#17989557—21no—
12C4×#11924037—21no—
13C4×#11924037—21no—
14C4×#11924037—21no—
15C4×#11924037C412yes—
16C4×#11924037C412yes—
17C2×#14556646—21no—
18C2×C4×#10596833C2212yes—
19C22×#11924012C411yescharacteristic
20C22×C4×Dic73C2211yescharacteristic
21M16×Dic73—21no—
22M16×Dic73—21no—
23M16×Dic73C412yes—
24M16×Dic73C2212yes—
25C2×#14556501—21no—
26C2×C8×Dic73—21no—
27C22×M16×C73C411yescharacteristic
28C2×#11924037—28no—
29C2×#11924037—22no—
30C2×#11924037—22no—
31C2×#11924037—22no—
32C22×#10596833—22no—
33#14556658—22no—
34C4⋊#10596833C2×C414yes—
35C22⋊#10596833—22no—
36C22⋊#10596833—22no—
37#14556646—22no—
38#14556646—22no—
39#14556646—22no—
40C4×#10596833—22no—
41C4×#10596833—22no—
42C4×#10596833C2×C414yes—
43C23×Dic73D411yescharacteristic
44C23×Dic73Q811yescharacteristic
45C2×#11924012—24no—
46C2×#11924012—21no—
47C2×#11924012C2×C412yes—
48C22×#10596831—21no—
49C2×C4×Dic73C2×C411yescharacteristic
50C2×C4×Dic73C2×C411yescharacteristic
51C2×C4×Dic73C2×C411yescharacteristic
52C2×C4×Dic73C2×C411yescharacteristic
53C2×C4×Dic73C2×C411yescharacteristic
54C2×C4×Dic73D411yescharacteristic
55C2×C4×Dic73D411yescharacteristic
56C2×C4×Dic73D411yescharacteristic
57C2×C4×Dic73D411yescharacteristic
58C2×C4×Dic73D411yescharacteristic
59C2×C4×Dic73Q811yescharacteristic
60C2×C4×Dic73C2311yescharacteristic
61#14556501—22no—
62#14556501—22no—
63C8×Dic73—22no—
64C8×Dic73—22no—
65C23×C4×C73C2×C411yescharacteristic
66C2×M16×C73—21no—
67C2×M16×C73—24no—
68C2×M16×C73C2×C412yes—
69C22×C8×C73—21no—
70#11924037—28no—
71#11924037—28no—
72#11924037—44no—
73C2×#10596833—28no—
74C2×#10596833—24no—
75C2×#10596833—24no—
76C22×Dic73—24no—
77C22×Dic73—42no—
78C22×Dic73C22⋊C411yescharacteristic
79C22×Dic73C22⋊C411yescharacteristic
80C22×Dic73C22⋊C411yescharacteristic
81C22×Dic73C22⋊C411yescharacteristic
82C22×Dic73C4⋊C411yescharacteristic
83C22×Dic73C4⋊C411yescharacteristic
84C22×Dic73C4⋊C411yescharacteristic
85C22×Dic73C4⋊C411yescharacteristic
86C22×Dic73C4⋊C411yescharacteristic
87C22×Dic73C4⋊C411yescharacteristic
88C22×Dic73C2×D411yescharacteristic
89C22×Dic73C2×Q811yescharacteristic
90#11924012—24no—
91#11924012—22no—
92#11924012M1614yescomplemented
93C2×#10596831—24no—
94C2×#10596831—22no—
95C4×Dic73C4211yescharacteristic
96C4×Dic73C4211yescharacteristic
97C4×Dic73C22⋊C411yescharacteristic
98C4×Dic73C22⋊C411yescharacteristic
99C4×Dic73C22⋊C411yescharacteristic
100C4×Dic73C22⋊C411yescharacteristic
101C4×Dic73C22⋊C411yescharacteristic
102C4×Dic73C22⋊C411yescharacteristic
103C4×Dic73C4⋊C411yescharacteristic
104C4×Dic73C4⋊C411yescharacteristic
105C4×Dic73C4⋊C411yescharacteristic
106C4×Dic73C4⋊C411yescharacteristic
107C4×Dic73C22×C411yescharacteristic
108C4×Dic73C22×C411yescharacteristic
109C4×Dic73C2×D411yescharacteristic
110C4×Dic73C2×D411yescharacteristic
111C24×C73C4⋊C411yescharacteristic
112C22×C4×C73—22no—
113C22×C4×C73—22no—
114C22×C4×C73C4211yescharacteristic
115C22×C4×C73C4211yescharacteristic
116C22×C4×C73C22⋊C411yescharacteristic
117C22×C4×C73C22⋊C411yescharacteristic
118C22×C4×C73C4⋊C411yescharacteristic
119C22×C4×C73C22×C411yescharacteristic
120M16×C73—24no—
121M16×C73—22no—
122M16×C73M1614yescomplemented
123C2×C8×C73—24no—
124C2×C8×C73—22no—
125C23×C73—22no—
126C23×C73—22no—
127C23×C73—41no—
128C23×C73#9511yescharacteristic
129C23×C73#9511yescharacteristic
130C23×C73C2×C4⋊C411yescharacteristic
131C2×C4×C73—22no—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73#9511yescharacteristic
139C2×C4×C73#9511yescharacteristic
140C2×C4×C73C2×C4211yescharacteristic
141C2×C4×C73C2×C22⋊C411yescharacteristic
142C2×C4×C73C2×M1612yes—
143C2×C4×C73C2×M1612yes—
144C2×Dic73—22no—
145C2×Dic73—22no—
146C2×Dic73—22no—
147C2×Dic73—22no—
148C2×Dic73—22no—
149C2×Dic73—22no—
150C2×Dic73—22no—
151C2×Dic73—22no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73#9511yescharacteristic
156C2×Dic73#9511yescharacteristic
157C2×Dic73#9511yescharacteristic
158C2×Dic73#9511yescharacteristic
159C2×Dic73#10414yes—
160C2×Dic73C2×C22⋊C411yescharacteristic
161C2×Dic73C2×C22⋊C411yescharacteristic
162C2×Dic73C2×C4⋊C411yescharacteristic
163C2×Dic73C2×C4⋊C411yescharacteristic
164#10596833—28no—
165#10596833—44no—
166C8×C73—24no—
167#10596831—24no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—44no—
176C22×C73#33912yes—
177C22×C73C2×#9511yescharacteristic
178C22×C73#42312yes—
179C4×C73—21no—
180C4×C73—21no—
181C4×C73C4×M1612yes—
182C4×C73#40711yescharacteristic
183C4×C73#40711yescharacteristic
184Dic73—21no—
185Dic73—21no—
186Dic73—21no—
187Dic73—21no—
188Dic73—21no—
189Dic73#33912yes—
190Dic73C2×#10412yes—
191Dic73#43012yes—
192#7887—731nomaximal, Sylow
193C2×C73—22no—
194C2×C73—22no—
195C2×C73—41no—
196C2×C73#172811yescharacteristic
197C2×C73C2×#33911yescharacteristic
198C2×C73#173811yescharacteristic
199#2920—731no—
200C2×C4×M16—731no—
201C2×C4×M16—731no—
202#1278—734no—
203C73#788711yescharacteristic, complemented, Sylow
204C22×M16—731no—
205C22×M16—1461no—
206C22×M16#6440811yescharacteristic
207C22×C42—731no—
208#432—732no—
209#432—1461no—
210#431—732no—
211#431—1461no—
212#423—732no—
213#407—1461no—
214C4×M16—732no—
215C4×M16—732no—
216C4×M16—732no—
217C4×M16—732no—
218C4×M16—1461no—
219C4×M16—1461no—
220C4×M16—1461no—
221C4×M16—1461no—
222C4×M16—1461no—
223C2×#97—1461no—
224C2×#97—1461no—
225C2×C4×C8—732no—
226C2×C4×C8—1461no—
227C23×C4—731no—
228C23×C4—731no—
229C23×C4C2×#6440811yescharacteristic
230C2×M16—21no—
231C2×M16—24no—
232C2×M16—732no—
233C2×M16—1462no—
234C2×M16—1464no—
235C2×M16—1462no—
236C2×M16—1462no—
237C2×M16—1468no—
238C2×M16—1461no—
239C2×M16C2×#6440812yes—
240C22×C8—21no—
241C22×C8—1462no—
242C22×C8—1461no—
243C2×C42—731no—
244C2×C42—731no—
245C2×C42—731no—
246C2×C42—731no—
247C2×C42—731no—
248C2×C42—731no—
249C2×C42—731no—
250C2×C42—731no—
251C2×C42—731no—
252C2×C42—731no—
253C2×C42—731no—
254C2×C42—731no—
255C4⋊C8—734no—
256C4⋊C8—1462no—
257C22⋊C8—1462no—
258C22⋊C8—1462no—
259#97—1462no—
260#97—1462no—
261#97—1462no—
262#97—1462no—
263#97—1462no—
264C4×C8—734no—
265C4×C8—1462no—
266C4×C8—1462no—
267C4×C8—1462no—
268C4×C8—1462no—
269C24C4⋊#6440811yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—2922no—
273C22×C4—731no—
274C22×C4—731no—
275C22×C4—731no—
276C22×C4—731no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—731no—
280C22×C4—731no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—731no—
284C22×C4—731no—
285C22×C4—1464no—
286C22×C4C4×#6440811yescharacteristic
287C22×C4C4×#6440811yescharacteristic
288C22×C4C22⋊#6440811yescharacteristic
289C22×C4C22⋊#6440811yescharacteristic
290C22×C4C4⋊#6440811yescharacteristic
291C22×C4C22×#6440811yescharacteristic
292M16—24no—
293M16—22no—
294M16—2924no—
295M16—734no—
296M16—1464no—
297M16—1468no—
298M16—1468no—
299M16—1462no—
300M16#1192403714yescomplemented
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—1464no—
304C2×C8—1464no—
305C2×C8—1464no—
306C2×C8—1468no—
307C2×C8—1462no—
308C42—731no—
309C42—731no—
310C42—731no—
311C42—731no—
312C42—731no—
313C42—731no—
314C42—731no—
315C42—731no—
316C42—731no—
317C42—731no—
318C42—731no—
319C42—731no—
320C42—731no—
321C42—731no—
322C42—731no—
323C42—731no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23(C2×#64408)⋊C411yescharacteristic
328C23(C2×#64408)⋊C411yescharacteristic
329C23C2×C4⋊#6440811yescharacteristic
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—2921no—
338C2×C4—2921no—
339C2×C4—2921no—
340C2×C4—734no—
341C2×C4—731no—
342C2×C4—731no—
343C2×C4—731no—
344C2×C4—731no—
345C2×C4—731no—
346C2×C4—731no—
347C2×C4—731no—
348C2×C4—731no—
349C2×C4—1462no—
350C2×C4—1462no—
351C2×C4—1462no—
352C2×C4—1462no—
353C2×C4—1462no—
354C2×C4—1462no—
355C2×C4—1462no—
356C2×C4—1462no—
357C2×C4(C2×#64408)⋊C411yescharacteristic
358C2×C4(C2×#64408)⋊C411yescharacteristic
359C2×C4C2×C4×#6440811yescharacteristic
360C2×C4C2×C22⋊#6440811yescharacteristic
361C2×C4C2×#1192403712yes—
362C2×C4D73⋊M1612yes—
363C8—24no—
364C8—2924no—
365C8—1468no—
366C8—1464no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—44no—
375C22#1798945912yes—
376C22C2×(C2×#64408)⋊C411yescharacteristic
377C22#1798959012yes—
378C4—21no—
379C4—21no—
380C4—732no—
381C4—732no—
382C4—732no—
383C4—1461no—
384C4—1461no—
385C4—1461no—
386C4—1461no—
387C4—1461no—
388C4C4×#1192403712yes—
389C4#1798955711yescharacteristic
390C4#1798955911yescharacteristic
391C2—22no—
392C2—22no—
393C2—41no—
394C2#2262559411yescharacteristic
395C2C2×#1798945911yescharacteristic
396C2#2262562411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
FittingC22×M16×C73C4
socleC22×C73C2×#95
prime radicalC24×C73C4⋊C4
cosocleC2×C4×Dic73C23
hypercenterC22×M16#64408
hypocenterC73#7887
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. #22624764C2semidirect
92. #11924012M16semidirect
122. M16×C73M16semidirect
203. C73#7887semidirect
300. M16#11924037semidirect

Maximal subgroups

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

subgroupindex
#226266072
C2×C4×#119240372
#226247642
C2×M16×Dic732
#788773