Finite Group Databasebeta

← #32978643

Subgroups — #32978643

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic73C211yesmaximal, characteristic
2#22609909C212yesmaximal
3C2×#17988313C211yesmaximal, characteristic
4#22607773C212yesmaximal
5C2×#324×C73C211yesmaximal, characteristic
6C22×C4⋊Dic73C411yescharacteristic
7C22×C4×Dic73C411yescharacteristic
8C22×C4×Dic73C411yescharacteristic
9C22×C4×Dic73C2211yescharacteristic
10C2×#14556520—22no—
11C2×C22⋊#10596831C2212yes—
12C4⋊C4×Dic73—22no—
13C4⋊C4×Dic73—22no—
14C2×#14556498C411yescharacteristic
15C2×#14556498C2211yescharacteristic
16#17988341—22no—
17#17988313—22no—
18C22×C4⋊C4×C73C2211yescharacteristic
19C2×C22⋊C8×C73C2212yes—
20#324×C73—22no—
21C23×Dic73C2×C411yescharacteristic
22C23×Dic73C2×C411yescharacteristic
23C23×Dic73C2×C411yescharacteristic
24C23×Dic73C2×C411yescharacteristic
25C22×#10596831—22no—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—22no—
29C2×C4×Dic73—21no—
30C2×C4×Dic73—21no—
31C2×C4×Dic73—22no—
32C2×C4×Dic73—21no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73—42no—
35C2×C4×Dic73—42no—
36C2×C4×Dic73C814yescomplemented
37C2×C4×Dic73D412yes—
38C2×C4×Dic73D412yes—
39#14556520—44no—
40C22⋊#10596831—22no—
41C22⋊#10596831—22no—
42#14556498—22no—
43#14556498—22no—
44C23×C4×C73C2×C411yescharacteristic
45C23×C4×C73C2×C411yescharacteristic
46C23×C4×C73C2311yescharacteristic
47C22×C8×C73—22no—
48C2×C4⋊C4×C73—22no—
49C2×C4⋊C4×C73—22no—
50C2×C4⋊C4×C73D412yes—
51C2×C4⋊C4×C73Q812yes—
52C22⋊C8×C73—22no—
53C22⋊C8×C73—22no—
54C22×Dic73—22no—
55C22×Dic73—22no—
56C22×Dic73—21no—
57C22×Dic73—21no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—41no—
63C22×Dic73—41no—
64C22×Dic73—41no—
65C22×Dic73—41no—
66C22×Dic73—41no—
67C22×Dic73—41no—
68C22×Dic73C22⋊C412yes—
69C22×Dic73C22⋊C411yescharacteristic
70C22×Dic73C22⋊C411yescharacteristic
71C22×Dic73C2×C812yes—
72C22×Dic73C2×C812yes—
73C22×Dic73M1612yes—
74C22×Dic73M1612yes—
75C2×#10596831—42no—
76C2×#10596831—42no—
77C2×#10596831—42no—
78C4⋊Dic73—28no—
79C4×Dic73—24no—
80C4×Dic73—24no—
81C4×Dic73—24no—
82C4×Dic73—42no—
83C4×Dic73—42no—
84C4×Dic73—42no—
85C24×C73C22×C411yescharacteristic
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—41no—
93C22×C4×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73—41no—
96C22×C4×C73C4⋊C412yes—
97C22×C4×C73C4⋊C412yes—
98C22×C4×C73C2×C812yes—
99C22×C4×C73M1612yes—
100C22×C4×C73C2×D412yes—
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73C2×Q811yescharacteristic
103C22×C4×C73#4112yes—
104C2×C8×C73—42no—
105C2×C8×C73—42no—
106C2×C8×C73—42no—
107C4⋊C4×C73—24no—
108C4⋊C4×C73—24no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73C2×C22⋊C411yescharacteristic
114C23×C73C2×C4⋊C411yescharacteristic
115C23×C73#11711yescharacteristic
116C23×C73C22×C811yescharacteristic
117C23×C73C2×M1611yescharacteristic
118C23×C73C2×M1611yescharacteristic
119C23×C73C2×M1611yescharacteristic
120C2×C4×C73—22no—
121C2×C4×C73—22no—
122C2×C4×C73—22no—
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—22no—
128C2×C4×C73—22no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—42no—
136C2×C4×C73C4⋊C814yescomplemented
137C2×C4×C73#12112yes—
138C2×C4×C73C22⋊Q812yes—
139C2×Dic73—21no—
140C2×Dic73—21no—
141C2×Dic73—24no—
142C2×Dic73—24no—
143C2×Dic73—22no—
144C2×Dic73—22no—
145C2×Dic73—22no—
146C2×Dic73—22no—
147C2×Dic73—22no—
148C2×Dic73—42no—
149C2×Dic73—42no—
150C2×Dic73—42no—
151C2×Dic73—42no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—42no—
157C2×Dic73—42no—
158C2×Dic73C22⋊C812yes—
159C2×Dic73C22⋊C812yes—
160C2×Dic73#9912yes—
161C2×Dic73#10112yes—
162C8×C73—82no—
163#10596831—82no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73#38011yescharacteristic
177C22×C73C2×C22⋊C811yescharacteristic
178C22×C73#40711yescharacteristic
179C22×C73C2×#9911yescharacteristic
180C22×C73#41011yescharacteristic
181C22×C73C2×#10111yescharacteristic
182C22×C73#41311yescharacteristic
183C22×C73C2×C4⋊C811yescharacteristic
184C22×C73#42311yescharacteristic
185C22×C73#43111yescharacteristic
186C22×C73#43211yescharacteristic
187C4×C73—24no—
188C4×C73—22no—
189C4×C73—22no—
190C4×C73—42no—
191Dic73—22no—
192Dic73—42no—
193Dic73—42no—
194Dic73—42no—
195Dic73#32414yescomplemented
196#8558—731nomaximal, Sylow
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—21no—
200C2×C73—21no—
201C2×C73C2×#32411yescharacteristic
202C2×C73#145811yescharacteristic
203C2×C73#146511yescharacteristic
204C2×C73#146711yescharacteristic
205C2×C73#179711yescharacteristic
206C2×C73#179911yescharacteristic
207C2×C73#180111yescharacteristic
208C2×C4×C4⋊C4—731no—
209#1797—732no—
210C2×#324—731no—
211C2×#324D7311yescharacteristic
212#1280—732no—
213C73#855811yescharacteristic, complemented, Sylow
214C22×C4⋊C4—731no—
215C22×C4⋊C4C2×D7311yescharacteristic
216C22×C42—731no—
217C22×C42—731no—
218C22×C42—731no—
219C2×C4⋊C8—1462no—
220C2×C22⋊C8—732no—
221C2×C22⋊C8C2×D7312yes—
222C4×C4⋊C4—1462no—
223C4×C4⋊C4—1462no—
224C2×#95—731no—
225C2×#95—731no—
226#336—1462no—
227#324—22no—
228#324—1462no—
229C23×C4—731no—
230C23×C4—731no—
231C23×C4—731no—
232C23×C4—731no—
233C23×C4C4×D7311yescharacteristic
234C23×C4C4×D7311yescharacteristic
235C23×C4C22×D7311yescharacteristic
236C22×C8—22no—
237C22×C8—1462no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1462no—
242C2×C4⋊C4—1462no—
243C2×C4⋊C4Dic14612yes—
244C2×C4⋊C4C22⋊D7312yes—
245C2×C42—2922no—
246C2×C42—2922no—
247C2×C42—732no—
248C2×C42—734no—
249C2×C42—732no—
250C2×C42—1461no—
251C2×C42—1461no—
252C2×C42—1461no—
253C2×C42—1461no—
254C2×C42—1462no—
255C4⋊C8—2924no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—1462no—
259C22⋊C8—1462no—
260#95—1462no—
261#95—1462no—
262C24C2×C4×D7311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—2921no—
274C22×C4—2921no—
275C22×C4—2921no—
276C22×C4—2921no—
277C22×C4—2921no—
278C22×C4—2921no—
279C22×C4—732no—
280C22×C4—732no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—732no—
284C22×C4—732no—
285C22×C4—732no—
286C22×C4—1461no—
287C22×C4—1461no—
288C22×C4—1461no—
289C22×C4—1461no—
290C22×C4—1461no—
291C22×C4—1461no—
292C22×C4—1462no—
293C22×C4—1462no—
294C22×C4#1192400712yes—
295C22×C4#1192400712yes—
296C22×C4C8×D7312yes—
297C22×C4#1192401112yes—
298C22×C4D4×D7312yes—
299C22×C4C2×C22⋊D7311yescharacteristic
300C22×C4C2×Dic14611yescharacteristic
301C22×C4#1192402712yes—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—2922no—
306C2×C8—2922no—
307C2×C8—2922no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—1468no—
311C42—2922no—
312C42—2922no—
313C42—2922no—
314C42—1464no—
315C42—1464no—
316C42—1464no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C22⋊C4×D7311yescharacteristic
322C23C2×#1192400711yescharacteristic
323C23#1455654711yescharacteristic
324C23C2×C8×D7311yescharacteristic
325C23M16×D7311yescharacteristic
326C23M16×D7311yescharacteristic
327C23C2×#1192401111yescharacteristic
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—2922no—
345C2×C4—2922no—
346C2×C4—2922no—
347C2×C4—2922no—
348C2×C4—2921no—
349C2×C4—2921no—
350C2×C4—2921no—
351C2×C4—2921no—
352C2×C4—2922no—
353C2×C4—2922no—
354C2×C4—732no—
355C2×C4—732no—
356C2×C4—732no—
357C2×C4—732no—
358C2×C4—1462no—
359C2×C4—1462no—
360C2×C4—1462no—
361C2×C4—1462no—
362C2×C4—1462no—
363C2×C4—1464no—
364C2×C4—1464no—
365C2×C4—1461no—
366C2×C4—1461no—
367C2×C4#1455652014yescomplemented
368C2×C4#1455656312yes—
369C2×C4#1455656812yes—
370C8—82no—
371C8—5842no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#1798844511yescharacteristic
385C22C22⋊C8×D7311yescharacteristic
386C22#1798854911yescharacteristic
387C22#99×D7311yescharacteristic
388C22#1798855911yescharacteristic
389C22#101×D7311yescharacteristic
390C22#1798857011yescharacteristic
391C22C2×#1455652011yescharacteristic
392C22#1798861011yescharacteristic
393C22#1798863711yescharacteristic
394C22#1798864111yescharacteristic
395C4—24no—
396C4—22no—
397C4—22no—
398C4—42no—
399C4—2922no—
400C4—2922no—
401C4—2922no—
402C4—734no—
403C4—1462no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2#324×D7311yescharacteristic
409C2#2260824411yescharacteristic
410C2#2260826811yescharacteristic
411C2#2260827311yescharacteristic
412C2#2260990911yescharacteristic
413C2#2260991711yescharacteristic
414C2#2260992511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC23×C73C22×C8
FrattiniC23×C4C22×D73
FittingC2×#324×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22×C4
cosocleC23×C4×C73C23
hypercenterC2×#324D73
hypocenterC73#8558
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
36. C2×C4×Dic73C8semidirect
136. C2×C4×C73C4⋊C8semidirect
195. Dic73#324semidirect
213. C73#8558semidirect
367. C2×C4#14556520semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic732
#226099092
C2×#179883132
#226077732
C2×#324×C732
#855873