Finite Group Databasebeta

← #34006164

Subgroups — #34006164

Order 18688.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989603C211yesmaximal, characteristic
2#22625821C211yesmaximal, characteristic
3C2×#17989432C212yesmaximal
4#22624769C212yesmaximal
5C2×#17988473C211yesmaximal, characteristic
6#17989603—24no—
7C2×C4⋊#10596833—21no—
8C2×C22⋊#10596833C2212yes—
9C2×C22⋊#10596833C2212yes—
10C2×C4×#10596833—21no—
11#17989453—21no—
12#17989453—21no—
13#17989432—24no—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C4×Dic73C2211yescharacteristic
16#17988473—21no—
17#17988473—21no—
18#17988473—21no—
19#17988473—21no—
20C2×#14556498C411yescharacteristic
21C2×#14556498C411yescharacteristic
22C2×#14556498C411yescharacteristic
23C2×#14556498C2211yescharacteristic
24C22×C4⋊C4×C73C411yescharacteristic
25C22×#10596833—22no—
26C22×#10596833—22no—
27C4⋊#10596833—24no—
28C22⋊#10596833—22no—
29C22⋊#10596833—22no—
30C22⋊#10596833—22no—
31C22⋊#10596833D414yes—
32C4×#10596833—24no—
33C23×Dic73C2×C411yescharacteristic
34C23×Dic73C2×C411yescharacteristic
35C23×Dic73C2×C411yescharacteristic
36C23×Dic73C2311yescharacteristic
37C2×C4⋊Dic73—22no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73—22no—
41C2×C4×Dic73—22no—
42C2×C4×Dic73—22no—
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73D411yescharacteristic
46C2×C4×Dic73D411yescharacteristic
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50#14556498—21no—
51#14556498—21no—
52#14556498—21no—
53#14556498—21no—
54#14556498—21no—
55C23×C4×C73C2×C411yescharacteristic
56C23×C4×C73C2×C411yescharacteristic
57C23×C4×C73C2×C411yescharacteristic
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—22no—
61C2×C4⋊C4×C73C814yescomplemented
62C2×#10596833—24no—
63C2×#10596833—42no—
64C2×#10596833—42no—
65C2×#10596833—42no—
66C2×#10596833—42no—
67C2×#10596833—42no—
68C22×Dic73—22no—
69C22×Dic73—22no—
70C22×Dic73—21no—
71C22×Dic73—21no—
72C22×Dic73—22no—
73C22×Dic73—22no—
74C22×Dic73—41no—
75C22×Dic73—41no—
76C22×Dic73—41no—
77C22×Dic73—41no—
78C22×Dic73—41no—
79C22×Dic73—41no—
80C22×Dic73—41no—
81C22×Dic73C22⋊C411yescharacteristic
82C22×Dic73C22⋊C411yescharacteristic
83C22×Dic73C2×D411yescharacteristic
84C22×Dic73C2×D411yescharacteristic
85C22×Dic73C2×D412yes—
86C22×Dic73#4112yes—
87C4⋊Dic73—41no—
88C4⋊Dic73—42no—
89C4⋊Dic73—41no—
90C4×Dic73—21no—
91C4×Dic73—21no—
92C4×Dic73—22no—
93C4×Dic73—22no—
94C4×Dic73—21no—
95C4×Dic73—21no—
96C24×C73C22×C411yescharacteristic
97C22×C4×C73—22no—
98C22×C4×C73—22no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73—22no—
102C22×C4×C73—21no—
103C22×C4×C73—21no—
104C22×C4×C73—22no—
105C22×C4×C73—41no—
106C22×C4×C73—41no—
107C22×C4×C73C22⋊C411yescharacteristic
108C22×C4×C73C22⋊C411yescharacteristic
109C22×C4×C73C2×C812yes—
110C22×C4×C73C2×C812yes—
111C22×C4×C73C2×C812yes—
112C22×C4×C73M1612yes—
113C4⋊C4×C73—24no—
114C4⋊C4×C73—24no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73C2×C22⋊C411yescharacteristic
120C23×C73C4×D412yes—
121C23×C73C22×C811yescharacteristic
122C23×C73C2×M1611yescharacteristic
123C23×C73#13112yes—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—24no—
129C2×C4×C73—24no—
130C2×C4×C73—22no—
131C2×C4×C73—22no—
132C2×C4×C73—24no—
133C2×C4×C73—42no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—42no—
139C2×C4×C73—42no—
140C2×C4×C73C22⋊C812yes—
141C2×C4×C73C22⋊C812yes—
142C2×Dic73—21no—
143C2×Dic73—21no—
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—21no—
149C2×Dic73—21no—
150C2×Dic73—21no—
151C2×Dic73—42no—
152C2×Dic73—42no—
153C2×Dic73—44no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—82no—
161C2×Dic73—82no—
162C2×Dic73#10112yes—
163C2×Dic73#12011yescharacteristic
164C2×Dic73#12112yes—
165C2×Dic73#12311yescharacteristic
166#10596833—44no—
167#10596833—82no—
168C22×C73—22no—
169C22×C73—22no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—22no—
175C22×C73—22no—
176C22×C73#38611yescharacteristic
177C22×C73C2×C22⋊C811yescharacteristic
178C22×C73#40811yescharacteristic
179C22×C73#41011yescharacteristic
180C22×C73#41011yescharacteristic
181C22×C73C2×#10111yescharacteristic
182C22×C73#41311yescharacteristic
183C22×C73C8×D412yes—
184C22×C73#43512yes—
185C4×C73—22no—
186C4×C73—22no—
187C4×C73—42no—
188C4×C73—42no—
189Dic73—21no—
190Dic73—22no—
191Dic73—21no—
192Dic73—22no—
193Dic73—41no—
194Dic73—81no—
195Dic73—81no—
196#9082—731nomaximal, Sylow
197C2×C73—22no—
198C2×C73—22no—
199C2×C73#145811yescharacteristic
200C2×C73#146111yescharacteristic
201C2×C73#147011yescharacteristic
202C2×C73#147311yescharacteristic
203C2×C73#186911yescharacteristic
204C2×C73#189911yescharacteristic
205C2×C73#190111yescharacteristic
206C2×#431—731no—
207C2×#387—731no—
208#1817—731no—
209C2×#324—732no—
210#1280—732no—
211C73#908211yescharacteristic, complemented, Sylow
212C22×C4⋊C4—731no—
213C22×C4⋊C4#6440811yescharacteristic
214C22×C42—731no—
215#431—1464no—
216C2×C4⋊C8—1461no—
217C2×C22⋊C8—732no—
218C2×C22⋊C8—732no—
219C2×C4×C8—1461no—
220#387—1461no—
221#387—1461no—
222#387—1461no—
223#387—1461no—
224C2×#95—731no—
225C2×#95—731no—
226C2×#95—731no—
227C2×#95—731no—
228#336—1461no—
229#336—1461no—
230#324—1464no—
231C23×C4—731no—
232C23×C4—731no—
233C23×C4—731no—
234C23×C4—731no—
235C23×C4C2×#6440811yescharacteristic
236C23×C4C2×#6440811yescharacteristic
237C23×C4C2×#6440811yescharacteristic
238C22×C8—1462no—
239C22×C8—1462no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—1462no—
244C2×C4⋊C4—1462no—
245C2×C4⋊C4—1461no—
246C2×C4⋊C4—1461no—
247C2×C4⋊C4#1059683314yescomplemented
248C2×C42—731no—
249C2×C42—731no—
250C2×C42—731no—
251C2×C42—731no—
252C2×C42—1462no—
253C2×C42—1462no—
254C4⋊C8—1464no—
255C22⋊C8—734no—
256C22⋊C8—1462no—
257C22⋊C8—1462no—
258C22⋊C8—1462no—
259C4×C8—1464no—
260#95—1461no—
261#95—1461no—
262#95—1461no—
263#95—1461no—
264#95—1461no—
265#95—1461no—
266#95—1461no—
267#95—1461no—
268C24C22×#6440811yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—2921no—
280C22×C4—2921no—
281C22×C4—2921no—
282C22×C4—2921no—
283C22×C4—2921no—
284C22×C4—2921no—
285C22×C4—2921no—
286C22×C4—732no—
287C22×C4—732no—
288C22×C4—731no—
289C22×C4—731no—
290C22×C4—731no—
291C22×C4—731no—
292C22×C4—1461no—
293C22×C4—1461no—
294C22×C4—1462no—
295C22×C4—1462no—
296C22×C4—1462no—
297C22×C4—1462no—
298C22×C4C22⋊#6440811yescharacteristic
299C22×C4C22⋊#6440811yescharacteristic
300C22×C4C2×#1059683312yes—
301C22×C4C2×#1059683312yes—
302C22×C4C2×#1059683312yes—
303C22×C4#1192403712yes—
304C2×C8—2922no—
305C2×C8—2922no—
306C2×C8—2922no—
307C2×C8—2922no—
308C2×C8—2922no—
309C2×C8—1464no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—2922no—
313C4⋊C4—2921no—
314C4⋊C4—2921no—
315C42—1461no—
316C42—1461no—
317C42—1462no—
318C42—1462no—
319C42—1461no—
320C42—1461no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23C2×C22⋊#6440811yescharacteristic
326C23D4×#6440812yes—
327C23C22×#1059683311yescharacteristic
328C23C2×#1192403711yescharacteristic
329C23#1455667812yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—2924no—
347C2×C4—2921no—
348C2×C4—2921no—
349C2×C4—2921no—
350C2×C4—2921no—
351C2×C4—2921no—
352C2×C4—2921no—
353C2×C4—2922no—
354C2×C4—2922no—
355C2×C4—5842no—
356C2×C4—5842no—
357C2×C4—732no—
358C2×C4—731no—
359C2×C4—731no—
360C2×C4—732no—
361C2×C4—1461no—
362C2×C4—1461no—
363C2×C4—1461no—
364C2×C4—1461no—
365C2×C4—1461no—
366C2×C4—1461no—
367C2×C4—1461no—
368C2×C4—1461no—
369C2×C4—1461no—
370C2×C4C22⋊#1059683312yes—
371C2×C4C22⋊#1059683312yes—
372C8—2924no—
373C8—5842no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22#1798953511yescharacteristic
383C22C2×C22⋊#1059683311yescharacteristic
384C22#1798956011yescharacteristic
385C22#1798956611yescharacteristic
386C22#1798956811yescharacteristic
387C22D73⋊#10111yescharacteristic
388C22#1798957311yescharacteristic
389C22D4×#1059683312yes—
390C22#1798961812yes—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—42no—
395C4—2921no—
396C4—5841no—
397C4—5841no—
398C4—1461no—
399C4—1462no—
400C4—1461no—
401C4—1462no—
402C2—22no—
403C2—22no—
404C2#2262516211yescharacteristic
405C2#2262517411yescharacteristic
406C2#2262520311yescharacteristic
407C2#2262521511yescharacteristic
408C2#2262590411yescharacteristic
409C2#2262595511yescharacteristic
410C2#2262595911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22×C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#9082
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
61. C2×C4⋊C4×C73C8semidirect
211. C73#9082semidirect
247. C2×C4⋊C4#10596833semidirect

Maximal subgroups

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

subgroupindex
C2×#179896032
#226258212
C2×#179894322
#226247692
C2×#179884732
#908273