Finite Group Databasebeta

← #34027603

Subgroups — #34027603

Order 18688.

876 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#22626052C212yesmaximal
2#22626005C214yesmaximal, complemented
3#22625846C211yesmaximal, characteristic, complemented
4#22625822C212yesmaximal, complemented
5#22625727C212yesmaximal, complemented
6C4×C4⋊#10596833C211yesmaximal, characteristic, complemented
7#22625657C212yesmaximal, complemented
8C4×D4×Dic73C211yesmaximal, characteristic
9C2×C4⋊#10596833C2212yes—
10C2×C4⋊#10596833C2212yes—
11C2×C22⋊#10596833C2214yes—
12C2×C22⋊#10596833C2214yes—
13C2×#14556646C2214yes—
14C2×C4×#10596833C2212yes—
15#17989453C2211yescharacteristic
16#17989453C2212yes—
17#17989453C2214yes—
18#17989453C2212yes—
19#17989453C2211yescharacteristic
20C2×D4×Dic73C411yescharacteristic
21C2×D4×Dic73C2211yescharacteristic
22C22×C4×Dic73C2212yes—
23C4⋊C4×Dic73C411yescharacteristic
24C4⋊C4×Dic73C2211yescharacteristic
25C4×C4⋊Dic73C411yescharacteristic
26C22⋊C4×Dic73C412yes—
27C22⋊C4×Dic73C2212yes—
28C4×C22⋊Dic73C412yes—
29C42×Dic73C2211yescharacteristic
30C2×C4×D4×C73C411yescharacteristic
31C22×#10596833C2312yes—
32C22×#10596833C2314yes—
33C22×#10596833C2312yes—
34C4⋊#10596833—24no—
35C4⋊#10596833—24no—
36C22⋊#10596833—28no—
37C22⋊#10596833—44no—
38#14556646—44no—
39C4×#10596833—24no—
40C23×Dic73C2×C412yes—
41C23×Dic73C2312yes—
42D4×Dic73—22no—
43D4×Dic73—22no—
44D4×Dic73—24no—
45C2×C4⋊Dic73C2×C411yescharacteristic
46C2×C4⋊Dic73C2×C411yescharacteristic
47C2×C22⋊Dic73C2×C412yes—
48C2×C22⋊Dic73C2×C412yes—
49C2×C4×Dic73—44no—
50C2×C4×Dic73C2×C411yescharacteristic
51C2×C4×Dic73C2×C411yescharacteristic
52C2×C4×Dic73C2×C411yescharacteristic
53C2×C4×Dic73C2×C412yes—
54C2×C4×Dic73C2×C412yes—
55C2×C4×Dic73C2311yescharacteristic
56C2×C4×Dic73C2311yescharacteristic
57C2×C4×Dic73C2312yes—
58C2×C4×Dic73C2311yescharacteristic
59#14556498C2×C412yes—
60#14556498C2×C412yes—
61#14556498C2×C412yes—
62C22×D4×C73C2×C411yescharacteristic
63C23×C4×C73C2×C412yes—
64C4×D4×C73—24no—
65C2×C4⋊C4×C73C2×C411yescharacteristic
66C2×C22⋊C4×C73C2×C412yes—
67C2×C42×C73C2×C411yescharacteristic
68C2×#10596833—22no—
69C2×#10596833—22no—
70C2×#10596833—22no—
71C2×#10596833—24no—
72C2×#10596833—22no—
73C2×#10596833—24no—
74C2×#10596833—24no—
75C2×#10596833—22no—
76C2×#10596833#4114yes—
77C22×Dic73—24no—
78C22×Dic73—24no—
79C22×Dic73—22no—
80C22×Dic73—22no—
81C22×Dic73—24no—
82C22×Dic73—24no—
83C22×Dic73—42no—
84C22×Dic73C22×C411yescharacteristic
85C22×Dic73C22×C412yes—
86C22×Dic73C22×C412yes—
87C22×Dic73C22×C411yescharacteristic
88C22×Dic73C22×C411yescharacteristic
89C22×Dic73C2411yescharacteristic
90C4⋊Dic73—22no—
91C4⋊Dic73—22no—
92C22⋊Dic73—24no—
93C22⋊Dic73—24no—
94C4×Dic73—22no—
95C4×Dic73—22no—
96C4×Dic73—22no—
97C4×Dic73—22no—
98C4×Dic73—24no—
99C4×Dic73—42no—
100C4×Dic73—42no—
101C4×Dic73—41no—
102C4×Dic73#4112yes—
103C4×Dic73#4112yes—
104C24×C73C22×C412yes—
105C2×D4×C73—22no—
106C2×D4×C73—22no—
107C2×D4×C73M1614yes—
108C22×C4×C73—24no—
109C22×C4×C73—42no—
110C22×C4×C73—42no—
111C22×C4×C73C22×C412yes—
112C22×C4×C73C22×C411yescharacteristic
113C22×C4×C73C22×C411yescharacteristic
114C22×C4×C73C22×C411yescharacteristic
115C4⋊C4×C73—22no—
116C22⋊C4×C73—24no—
117C42×C73M1614yes—
118C23×C73—22no—
119C23×C73—22no—
120C23×C73—22no—
121C23×C73—22no—
122C23×C73—22no—
123C23×C73—22no—
124C23×C73C2×M1614yes—
125C23×C73C23×C411yescharacteristic
126D4×C73—28no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—22no—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—48no—
135C2×C4×C73#11712yes—
136C2×C4×C73C2×M1612yes—
137C2×C4×C73C2×M1612yes—
138C2×C4×C73C2×M1612yes—
139C2×C4×C73C2×M1612yes—
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—21no—
143C2×Dic73—21no—
144C2×Dic73—22no—
145C2×Dic73—21no—
146C2×Dic73—22no—
147C2×Dic73—22no—
148C2×Dic73—22no—
149C2×Dic73—21no—
150C2×Dic73—22no—
151C2×Dic73—22no—
152C2×Dic73—21no—
153C2×Dic73—48no—
154C2×Dic73—44no—
155C2×Dic73—44no—
156C2×Dic73#11712yes—
157C2×Dic73#13112yes—
158C2×Dic73C2×#4111yescharacteristic
159C2×Dic73C2×#4112yes—
160C2×Dic73C2×#4111yescharacteristic
161C2×Dic73H(5,𝔽2)12yes—
162#10596833—24no—
163#10596833—42no—
164#10596833—44no—
165C22×C73—24no—
166C22×C73—24no—
167C22×C73—24no—
168C22×C73—42no—
169C22×C73—42no—
170C22×C73—42no—
171C22×C73—42no—
172C22×C73C2×#11711yescharacteristic
173C22×C73C4×#4111yescharacteristic
174C22×C73#51811yescharacteristic
175C22×C73C22×M1611yescharacteristic
176C22×C73C22×M1611yescharacteristic
177C22×C73C2×#13111yescharacteristic
178C22×C73#56811yescharacteristic
179C4×C73—22no—
180C4×C73—22no—
181C4×C73—42no—
182C4×C73#43214yes—
183Dic73—22no—
184Dic73—22no—
185Dic73—42no—
186Dic73—41no—
187Dic73—42no—
188Dic73#55312yes—
189Dic73#55912yes—
190#22839—731nomaximal, Sylow
191C2×C73—44no—
192C2×C73#235711yescharacteristic
193C2×C73C2×#43211yescharacteristic
194C2×C73#292511yescharacteristic
195C2×C73#296511yescharacteristic
196C2×C73#297011yescharacteristic
197C2×C73#297411yescharacteristic
198C2×C73#297511yescharacteristic
199C42×D4—731no—
200#1954—732no—
201#1924—734no—
202#1831—731no—
203#1817—732no—
204#1770—732no—
205C4×C4⋊C8—731no—
206#1749—732no—
207C73#2283911yescharacteristic, complemented, Sylow
208C2×C4×D4—731no—
209C2×C4×D4—731no—
210C2×C4×D4#6440811yescharacteristic
211C22×C42—732no—
212C2×C4⋊C8—732no—
213C2×C4⋊C8—732no—
214C2×C22⋊C8—734no—
215C2×C22⋊C8—734no—
216C2×#97—734no—
217C2×C4×C8—732no—
218C4×C4⋊C4—731no—
219C4×C4⋊C4—731no—
220C4×C4⋊C4—731no—
221C4×C22⋊C4—732no—
222C4×C22⋊C4—732no—
223C4×C22⋊C4—732no—
224C43—731no—
225#336—731no—
226#336—732no—
227#336—734no—
228#336—732no—
229#336—731no—
230C22×D4C2×#6440811yescharacteristic
231C23×C4—732no—
232C23×C4—732no—
233C23×C4C2×#6440812yes—
234C22×C8—732no—
235C22×C8—732no—
236C22×C8—734no—
237C4×D4—24no—
238C4×D4—1462no—
239C4×D4—1462no—
240C4×D4—1464no—
241C2×C4⋊C4—731no—
242C2×C4⋊C4—731no—
243C2×C4⋊C4C2×#6440811yescharacteristic
244C2×C22⋊C4—732no—
245C2×C22⋊C4—732no—
246C2×C22⋊C4C2×#6440812yes—
247C2×C42—2924no—
248C2×C42—731no—
249C2×C42—732no—
250C2×C42—732no—
251C2×C42—731no—
252C2×C42—731no—
253C2×C42—731no—
254C2×C42—731no—
255C2×C42—732no—
256C2×C42—731no—
257C2×C42C2×#6440811yescharacteristic
258C4⋊C8—1464no—
259C4⋊C8—1464no—
260C22⋊C8—2924no—
261C22⋊C8—1468no—
262#97—2924no—
263C4×C8—1464no—
264#95—732no—
265#95—732no—
266#95—732no—
267C24C22×#6440812yes—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4#1192403814yes—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—2922no—
275C22×C4—731no—
276C22×C4—731no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—732no—
280C22×C4—732no—
281C22×C4—1464no—
282C22×C4—1464no—
283C22×C4—1462no—
284C22×C4—1462no—
285C22×C4—1464no—
286C22×C4—1464no—
287C22×C4C22×#6440812yes—
288C22×C4C22×#6440811yescharacteristic
289C22×C4C22×#6440811yescharacteristic
290C22×C4C22×#6440811yescharacteristic
291C2×C8—734no—
292C2×C8—1462no—
293C2×C8—1464no—
294C2×C8—1462no—
295C2×C8—1464no—
296C2×C8—1464no—
297C2×C8—1462no—
298C2×C8—1462no—
299C2×C8—1462no—
300C4⋊C4—22no—
301C4⋊C4—1462no—
302C4⋊C4—1462no—
303C22⋊C4—24no—
304C22⋊C4—1464no—
305C22⋊C4—1464no—
306C42—2922no—
307C42—2922no—
308C42—2921no—
309C42—732no—
310C42—732no—
311C42—1464no—
312C42—1462no—
313C42—1462no—
314C42—1462no—
315C42—1462no—
316C42#1192403714yes—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23C2×#1192403814yes—
324C23C23×#6440811yescharacteristic
325D4—28no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—48no—
334C2×C4—2924no—
335C2×C4—2924no—
336C2×C4—2928no—
337C2×C4—731no—
338C2×C4—731no—
339C2×C4—732no—
340C2×C4—732no—
341C2×C4—732no—
342C2×C4—732no—
343C2×C4—1462no—
344C2×C4—1461no—
345C2×C4—1462no—
346C2×C4—1462no—
347C2×C4—1462no—
348C2×C4—1461no—
349C2×C4—1462no—
350C2×C4—1462no—
351C2×C4—1461no—
352C2×C4—1461no—
353C2×C4—1461no—
354C2×C4—1461no—
355C2×C4—1461no—
356C2×C4#1455667012yes—
357C2×C4C2×#1192403712yes—
358C2×C4C2×#1192403712yes—
359C2×C4C2×#1192403712yes—
360C2×C4C2×#1192403812yes—
361C8—2924no—
362C8—2922no—
363C8—1464no—
364C22—24no—
365C22—24no—
366C22—24no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22C2×#1455667011yescharacteristic
372C22#41×#6440811yescharacteristic
373C22#1798965411yescharacteristic
374C22C22×#1192403711yescharacteristic
375C22C22×#1192403811yescharacteristic
376C22D73⋊#13111yescharacteristic
377C22#1798966511yescharacteristic
378C4—22no—
379C4—22no—
380C4—42no—
381C4—2922no—
382C4—2921no—
383C4—2922no—
384C4—732no—
385C4—732no—
386C4—1462no—
387C4—1462no—
388C4#1798960814yes—
389C2—44no—
390C2#2262645211yescharacteristic
391C2C2×#1798960811yescharacteristic
392C2#2262662811yescharacteristic
393C2#2262674011yescharacteristic
394C2#2262676911yescharacteristic
395C2#2262679111yescharacteristic
396C2#2262679311yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×C4×D4×C73C4
prime radicalC22×D4×C73C2×C4
cosocleC22×Dic73C24
hypercenterC2×C4×D4#64408
hypocenterC73#22839
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #22626005C2semidirect
3. #22625846C2semidirect
4. #22625822C2semidirect
5. #22625727C2semidirect
6. C4×C4⋊#10596833C2semidirect
7. #22625657C2semidirect
207. C73#22839semidirect

Maximal subgroups

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

subgroupindex
#226260522
#226260052
#226258462
#226258222
#226257272
C4×C4⋊#105968332
#226256572
C4×D4×Dic732
#2283973