Finite Group Databasebeta

← #34001324

Subgroups — #34001324

Order 18688.

596 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556648C211yesmaximal, characteristic
2#22625592C211yesmaximal, characteristic
3#22624752C212yesmaximal, complemented
4#22624749C212yesmaximal, complemented
5C4×#14556571C211yesmaximal, characteristic
6C22×#11924036—21no—
7C2×#14556648—21no—
8C2×C4×#10596833—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×#14556571C411yescharacteristic
13C2×C4×Dic146C411yescharacteristic
14C2×C42×D73C2211yescharacteristic
15#17988492C2211yescharacteristic
16#17988486C2211yescharacteristic
17#17988472—21no—
18#17988461—21no—
19#17988444C411yescharacteristic
20C4×C4⋊Dic73—21no—
21C4×#11924005—21no—
22C4×C4⋊C4×C73C411yescharacteristic
23C2×#11924036—24no—
24C2×#11924036—42no—
25C2×#11924036—42no—
26C22×#10596833—22no—
27C22×#10596833—22no—
28#14556648—24no—
29C4×#10596833—42no—
30C22×Dic146C2×C411yescharacteristic
31C22×C4×D73C2×C411yescharacteristic
32C22×C4×D73D411yescharacteristic
33C22×C4×D73D411yescharacteristic
34#14556571—22no—
35#14556571C814yescomplemented
36C4×Dic146—42no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73C2×C411yescharacteristic
40C2×#11924007—21no—
41C2×#11924005—21no—
42C2×#11924005—21no—
43C2×C4×Dic73—21no—
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73D411yescharacteristic
46C2×C4×Dic73C2311yescharacteristic
47C42×D73—42no—
48#14556498—21no—
49#14556498—21no—
50#14556498—21no—
51#14556498—21no—
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C4⋊C4×C73C2×C411yescharacteristic
54C2×C42×C73—21no—
55C2×C42×C73C2×C411yescharacteristic
56#95×C73—21no—
57#11924036—48no—
58C2×#10596833—24no—
59C2×#10596833—24no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C2×#10596833—42no—
63C2×#10596833—42no—
64C23×D73C22⋊C411yescharacteristic
65C2×Dic146—21no—
66C2×Dic146—41no—
67C2×Dic146—41no—
68C2×Dic146C2×C812yescomplemented
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73C22⋊C411yescharacteristic
72C22×Dic73C2×D411yescharacteristic
73C22×Dic73C2×D411yescharacteristic
74C2×C4×D73—22no—
75C2×C4×D73—22no—
76C2×C4×D73—22no—
77C2×C4×D73—22no—
78C2×C4×D73—41no—
79C2×C4×D73—41no—
80C2×C4×D73—41no—
81C2×C4×D73C2×C812yes—
82C2×C4×D73M1612yes—
83C2×C4×D73SD1612yes—
84C2×C4×D73Q1612yes—
85C4⋊Dic73—24no—
86C4⋊Dic73—42no—
87C4⋊Dic73—41no—
88#11924007—42no—
89#11924005—24no—
90#11924005—42no—
91C4×Dic73—22no—
92C4×Dic73—41no—
93C4×Dic73—41no—
94C4×Dic73—42no—
95C22×C4×C73—21no—
96C22×C4×C73—21no—
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22×C411yescharacteristic
100C4⋊C4×C73—21no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73C2×C812yescomplemented
103C42×C73—22no—
104C42×C73—42no—
105C23×C73C2×C22⋊C411yescharacteristic
106C22×D73—21no—
107C22×D73—21no—
108C22×D73—21no—
109C22×D73—21no—
110C22×D73C22⋊C812yes—
111C22×D73#10312yes—
112C22×D73#10412yes—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—22no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73#9912yes—
125C2×C4×C73#10312yes—
126C2×C4×C73#10412yes—
127C2×C4×C73C22×C811yescharacteristic
128C2×C4×C73C2×M1611yescharacteristic
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—21no—
134C2×Dic73—21no—
135C2×Dic73—22no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73C22⋊C812yes—
142C2×Dic73C2×SD1611yescharacteristic
143C2×Dic73C2×Q1611yescharacteristic
144C2×Dic73#13512yes—
145C4×D73—24no—
146C4×D73—24no—
147C4×D73—42no—
148C4×D73—42no—
149C4×D73—44no—
150Dic146—42no—
151Dic146—42no—
152#10596833—44no—
153#10596833—44no—
154C22×C73C2×C22⋊C811yescharacteristic
155C22×C73C2×#9911yescharacteristic
156C22×C73#41311yescharacteristic
157C22×C73C2×#10311yescharacteristic
158C22×C73#41611yescharacteristic
159C22×C73C2×#10411yescharacteristic
160C22×C73C2×#10411yescharacteristic
161C2×D73—22no—
162C2×D73—22no—
163C2×D73—22no—
164C2×D73—22no—
165C2×D73—22no—
166C2×D73—22no—
167C2×D73Q8⋊C814yes—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—42no—
171C4×C73—41no—
172C4×C73—81no—
173C4×C73#32814yes—
174Dic73—22no—
175Dic73—21no—
176Dic73—21no—
177Dic73—22no—
178Dic73—42no—
179Dic73—81no—
180#7566—731nomaximal, Sylow
181C2×C73#145911yescharacteristic
182C2×C73C2×Q8⋊C811yescharacteristic
183C2×C73#147611yescharacteristic
184C2×C73C2×#32811yescharacteristic
185C2×C73#150111yescharacteristic
186C2×C73#152411yescharacteristic
187C2×C73#154211yescharacteristic
188D73—24no—
189C4×C22⋊Q8—731no—
190C4×C22⋊C8—731no—
191C2×#336—731no—
192#1272—732no—
193#1271—732no—
194C73#756611yescharacteristic, complemented, Sylow
195C23×C8—1461no—
196C2×C22⋊Q8—731no—
197C2×C4×Q8—731no—
198C22×C42—731no—
199C2×C22⋊C8—1461no—
200C2×C4×C8—1461no—
201#391—731no—
202#389—731no—
203#387—1461no—
204#385—1461no—
205#380—731no—
206C4×C4⋊C4—1461no—
207C4×C4⋊C4#6440811yescharacteristic
208C4×C22⋊C4—1461no—
209#336—732no—
210#336—732no—
211#336—1461no—
212C22×Q8—731no—
213C23×C4—731no—
214C23×C4—731no—
215C23×C4—731no—
216C22×C8—2922no—
217C22×C8—2922no—
218C22×C8—1462no—
219C22×C8—1462no—
220C22×C8—1464no—
221C22⋊Q8—734no—
222C22⋊Q8—1462no—
223C4×Q8—2922no—
224C2×C4⋊C4—731no—
225C2×C4⋊C4—1461no—
226C2×C4⋊C4—1461no—
227C2×C4⋊C4—1461no—
228C2×C4⋊C4C2×#6440811yescharacteristic
229C2×C4⋊C4C2×#6440811yescharacteristic
230C2×C22⋊C4—1461no—
231C2×C22⋊C4—1461no—
232C2×C42—21no—
233C2×C42—2922no—
234C2×C42—731no—
235C2×C42—731no—
236C2×C42—731no—
237C2×C42—1461no—
238C2×C42C2×#6440811yescharacteristic
239C22⋊C8—1464no—
240C4×C8—2922no—
241#95—21no—
242#95—1461no—
243#95—1461no—
244#95—1461no—
245#95—1461no—
246C24—731no—
247C2×Q8—2921no—
248C2×Q8—2921no—
249C2×Q8—732no—
250C2×Q8—1461no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—2921no—
254C22×C4—2921no—
255C22×C4—2921no—
256C22×C4—732no—
257C22×C4—732no—
258C22×C4—731no—
259C22×C4—731no—
260C22×C4—731no—
261C22×C4—732no—
262C22×C4—732no—
263C22×C4—1462no—
264C22×C4—1462no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4—1462no—
268C22×C4—1462no—
269C22×C4C22⋊#6440811yescharacteristic
270C22×C4C22⋊#6440811yescharacteristic
271C22×C4C22×#6440811yescharacteristic
272C2×C8—2922no—
273C2×C8—2922no—
274C2×C8—2922no—
275C2×C8—2922no—
276C2×C8—2928no—
277C2×C8—1464no—
278C2×C8—1464no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—2921no—
282C4⋊C4—2922no—
283C4⋊C4—2922no—
284C4⋊C4—1464no—
285C4⋊C4#1192403612yescomplemented
286C22⋊C4—2922no—
287C22⋊C4—1464no—
288C42—22no—
289C42—42no—
290C42—2921no—
291C42—2921no—
292C42—2922no—
293C42—1462no—
294C23—732no—
295C23—732no—
296C23—732no—
297C23—1461no—
298C23—1461no—
299C23—1461no—
300C23—1461no—
301C23C2×C22⋊#6440811yescharacteristic
302Q8—2922no—
303Q8—2922no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—2922no—
316C2×C4—2922no—
317C2×C4—2924no—
318C2×C4—2921no—
319C2×C4—2921no—
320C2×C4—2921no—
321C2×C4—2921no—
322C2×C4—2921no—
323C2×C4—732no—
324C2×C4—731no—
325C2×C4—731no—
326C2×C4—732no—
327C2×C4—1462no—
328C2×C4—1461no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4—1461no—
333C2×C4—1461no—
334C2×C4—1464no—
335C2×C4—1464no—
336C2×C4#1455665012yes—
337C2×C4#1455665412yes—
338C2×C4#1455665512yes—
339C2×C4C2×#1192403611yescharacteristic
340C2×C4D73⋊M1611yescharacteristic
341C8—2924no—
342C8—2924no—
343C22—734no—
344C22—1462no—
345C22—1462no—
346C22—1462no—
347C22—1462no—
348C22—1462no—
349C22—1462no—
350C22#1798955511yescharacteristic
351C22C2×#1455665011yescharacteristic
352C22#1798957311yescharacteristic
353C22C2×#1455665411yescharacteristic
354C22#1798957711yescharacteristic
355C22C2×#1455665511yescharacteristic
356C22D73⋊#10411yescharacteristic
357C4—21no—
358C4—21no—
359C4—41no—
360C4—42no—
361C4—81no—
362C4—2922no—
363C4—5841no—
364C4—1461no—
365C4—1461no—
366C4—1462no—
367C4—1462no—
368C4#1798944014yes—
369C2—1464no—
370C2#2262516811yescharacteristic
371C2#2262522011yescharacteristic
372C2#2262522311yescharacteristic
373C2C2×#1798944011yescharacteristic
374C2#2262530811yescharacteristic
375C2#2262538111yescharacteristic
376C2#2262543211yescharacteristic
subgroupG/H
centerC22×C4C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC4×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC4×C4⋊C4#64408
hypocenterC73#7566
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22624752C2semidirect
4. #22624749C2semidirect
35. #14556571C8semidirect
68. C2×Dic146C2×C8semidirect
102. C4⋊C4×C73C2×C8semidirect
194. C73#7566semidirect
285. C4⋊C4#11924036semidirect

Maximal subgroups

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

subgroupindex
C4×#145566482
#226255922
#226247522
#226247492
C4×#145565712
#756673