Finite Group Databasebeta

← #34026716

Subgroups — #34026716

Order 18688.

780 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989612C212yesmaximal
2#22626033C212yesmaximal, complemented
3#22625988C212yesmaximal, complemented
4#22625987C212yesmaximal, complemented
5#22625825C212yesmaximal
6#22625788C211yesmaximal, characteristic
7#22625731C212yesmaximal, complemented
8#22625658C211yesmaximal, characteristic
9#22612336C211yesmaximal, characteristic
10#17989612—28no—
11C2×#14556658C2212yes—
12C2×#14556658C2212yes—
13C2×C4⋊#10596833C2212yes—
14C2×#14556648C2211yescharacteristic
15C2×#14556648C2212yes—
16C2×#14556648C2211yescharacteristic
17C2×#14556646C2212yes—
18C2×#14556646C2212yes—
19C2×C4×#10596833C2212yes—
20#17989453C2212yes—
21#17989453C2212yes—
22#17989453C2212yes—
23#17989453C2212yes—
24#17989453C2212yes—
25#17989453C2211yescharacteristic
26#17989453C2211yescharacteristic
27C2×#14556549C2212yes—
28C2×C42×D73C2211yescharacteristic
29#17988457C411yescharacteristic
30#17988452C412yes—
31#17988444C412yes—
32#17988444C411yescharacteristic
33#17988444C411yescharacteristic
34C4⋊C4×Dic73C2211yescharacteristic
35C4⋊C4×Dic73C2212yes—
36C4⋊C4×Dic73C2211yescharacteristic
37#384×C73C411yescharacteristic
38C22×#10596833C2312yes—
39C22×#10596833C2312yes—
40C22×#10596833C2312yes—
41C22×#10596833C2312yes—
42#14556658—24no—
43#14556658—24no—
44C4⋊#10596833—24no—
45#14556648—24no—
46#14556648—22no—
47#14556648D414yes—
48#14556646—24no—
49#14556646—24no—
50C4×#10596833—24no—
51C22×C4×D73C2×C411yescharacteristic
52C22×C4×D73C2×C411yescharacteristic
53C22×C4×D73C2311yescharacteristic
54#14556549—24no—
55#14556549—24no—
56C2×C4⋊Dic73C2×C412yes—
57C2×C4⋊Dic73C2×C412yes—
58C2×C4⋊Dic73C2×C411yescharacteristic
59C2×C4⋊Dic73C2×C411yescharacteristic
60C2×#11924005C2×C412yes—
61C2×#11924005C2×C412yes—
62C2×C4×Dic73C2×C411yescharacteristic
63C2×C4×Dic73C2311yescharacteristic
64C2×C4×Dic73C2312yes—
65C2×C4×Dic73C2312yes—
66C2×C4×Dic73C2311yescharacteristic
67C42×D73—24no—
68#14556498C2×C412yes—
69#14556498C2×C412yes—
70#14556498C2×C412yes—
71#14556498C2×C412yes—
72C2×C4⋊C4×C73C2×C412yes—
73C2×C4⋊C4×C73C2×C411yescharacteristic
74C2×C4⋊C4×C73C2×C411yescharacteristic
75C2×C4⋊C4×C73C2×C412yes—
76C2×C42×C73C2×C411yescharacteristic
77C2×#10596833—22no—
78C2×#10596833—22no—
79C2×#10596833—22no—
80C2×#10596833—22no—
81C2×#10596833—22no—
82C2×#10596833—22no—
83C2×#10596833—22no—
84C2×#10596833—22no—
85C2×#10596833—22no—
86C2×#10596833—22no—
87C2×#10596833—22no—
88C2×#10596833C2×D414yes—
89C23×D73C22×C411yescharacteristic
90C22×Dic73C22×C411yescharacteristic
91C22×Dic73C22×C412yes—
92C22×Dic73C22×C412yes—
93C22×Dic73C22×C411yescharacteristic
94C22×Dic73C2411yescharacteristic
95C2×C4×D73—22no—
96C2×C4×D73—22no—
97C2×C4×D73—22no—
98C2×C4×D73—22no—
99C2×C4×D73—22no—
100C2×C4×D73—41no—
101C2×C4×D73—41no—
102C2×C4×D73—41no—
103C2×C4×D73C2×D412yes—
104C2×C4×D73#4112yes—
105C4⋊Dic73—24no—
106C4⋊Dic73—24no—
107C4⋊Dic73—41no—
108C4⋊Dic73—41no—
109#11924005—24no—
110#11924005—24no—
111C4×Dic73—22no—
112C4×Dic73—22no—
113C4×Dic73—22no—
114C4×Dic73—22no—
115C4×Dic73—22no—
116C4×Dic73—22no—
117C4×Dic73#4114yes—
118C22×C4×C73C22×C412yes—
119C22×C4×C73C22×C412yes—
120C22×C4×C73C22×C411yescharacteristic
121C22×C4×C73C22×C411yescharacteristic
122C22×C4×C73C22×C411yescharacteristic
123C4⋊C4×C73—24no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—42no—
127C42×C73—22no—
128C23×C73C23×C411yescharacteristic
129C22×D73—21no—
130C22×D73—21no—
131C22×D73—21no—
132C22×D73—21no—
133C22×D73—21no—
134C22×D73—21no—
135C22×D73C4×D412yes—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—22no—
143C2×C4×C73—22no—
144C2×C4×C73—22no—
145C2×C4×C73—22no—
146C2×C4×C73—22no—
147C2×C4×C73—22no—
148C2×C4×C73#11712yes—
149C2×C4×C73C4×D412yes—
150C2×C4×C73#13112yes—
151C2×Dic73—21no—
152C2×Dic73—21no—
153C2×Dic73—21no—
154C2×Dic73—22no—
155C2×Dic73—22no—
156C2×Dic73—22no—
157C2×Dic73—22no—
158C2×Dic73—22no—
159C2×Dic73—22no—
160C2×Dic73—21no—
161C2×Dic73—21no—
162C2×Dic73#11712yes—
163C2×Dic73C22×D411yescharacteristic
164C2×Dic73C2×#4112yes—
165C2×Dic73C2×#4111yescharacteristic
166C2×Dic732−1+412yes—
167C4×D73—24no—
168C4×D73—24no—
169C4×D73—44no—
170C4×D73—44no—
171#10596833—24no—
172#10596833—42no—
173#10596833—42no—
174#10596833—42no—
175C22×C73C2×#11711yescharacteristic
176C22×C73C2×C4×D411yescharacteristic
177C22×C73#51911yescharacteristic
178C22×C73C2×#13111yescharacteristic
179C22×C73#56811yescharacteristic
180C22×C73#56811yescharacteristic
181C22×C73#56811yescharacteristic
182C2×D73—22no—
183C2×D73—22no—
184C2×D73—22no—
185C2×D73—41no—
186C2×D73—41no—
187C2×D73—41no—
188C2×D73—41no—
189C4×C73—22no—
190C4×C73—22no—
191C4×C73—22no—
192C4×C73—42no—
193C4×C73—42no—
194Dic73—22no—
195Dic73—41no—
196Dic73—42no—
197Dic73—42no—
198Dic73#54712yes—
199Dic73#55212yes—
200#22602—731nomaximal, Sylow
201C2×C73#234411yescharacteristic
202C2×C73#292111yescharacteristic
203C2×C73#292411yescharacteristic
204C2×C73#293211yescharacteristic
205C2×C73#293511yescharacteristic
206C2×C73#298511yescharacteristic
207C2×C73#298711yescharacteristic
208D73—42no—
209C2×#433—732no—
210#2285—731no—
211#1940—732no—
212#1917—732no—
213#1916—732no—
214#1818—732no—
215#1797—731no—
216#1771—732no—
217#1749—731no—
218C73#2260211yescharacteristic, complemented, Sylow
219C2×#117—732no—
220C22×C42—731no—
221#433—1468no—
222C2×C4⋊C8—732no—
223C2×C4⋊C8—732no—
224C2×C4⋊C8—732no—
225C2×C22⋊C8—731no—
226C2×C22⋊C8—732no—
227C2×C22⋊C8—731no—
228C2×#97—732no—
229C2×#97—732no—
230C2×C4×C8—732no—
231#384—731no—
232#384#6440811yescharacteristic
233#382—732no—
234#380—731no—
235#380—731no—
236#380—732no—
237C4×C4⋊C4—732no—
238C4×C4⋊C4—731no—
239C4×C4⋊C4—731no—
240#336—731no—
241#336—731no—
242#336—732no—
243#336—732no—
244#336—732no—
245#336—732no—
246#336—732no—
247C23×C4—731no—
248C23×C4—731no—
249C23×C4—731no—
250C22×C8—732no—
251C22×C8—732no—
252C22×C8—732no—
253C22×C8—732no—
254#117—1464no—
255#117—1464no—
256C2×C4⋊C4—731no—
257C2×C4⋊C4—731no—
258C2×C4⋊C4—732no—
259C2×C4⋊C4—732no—
260C2×C4⋊C4C2×#6440812yes—
261C2×C4⋊C4C2×#6440812yes—
262C2×C4⋊C4C2×#6440811yescharacteristic
263C2×C4⋊C4C2×#6440811yescharacteristic
264C2×C22⋊C4—732no—
265C2×C22⋊C4—732no—
266C2×C42—731no—
267C2×C42—732no—
268C2×C42—732no—
269C2×C42—731no—
270C2×C42—731no—
271C2×C42—1464no—
272C2×C42C2×#6440811yescharacteristic
273C4⋊C8—1464no—
274C4⋊C8—1464no—
275C4⋊C8—1464no—
276C22⋊C8—734no—
277C22⋊C8—1464no—
278C22⋊C8—1462no—
279#97—1464no—
280#97—1464no—
281C4×C8—1464no—
282#95—732no—
283#95—732no—
284#95—732no—
285#95—732no—
286C24—731no—
287C22×C4—2921no—
288C22×C4—2921no—
289C22×C4—2921no—
290C22×C4—731no—
291C22×C4—731no—
292C22×C4—732no—
293C22×C4—732no—
294C22×C4—731no—
295C22×C4—732no—
296C22×C4—732no—
297C22×C4—1462no—
298C22×C4—1462no—
299C22×C4—1462no—
300C22×C4—1462no—
301C22×C4—1462no—
302C22×C4C22×#6440812yes—
303C22×C4C22×#6440812yes—
304C22×C4C22×#6440811yescharacteristic
305C22×C4C22×#6440811yescharacteristic
306C22×C4C22×#6440811yescharacteristic
307C2×C8—734no—
308C2×C8—1462no—
309C2×C8—1462no—
310C2×C8—1462no—
311C2×C8—1462no—
312C2×C8—1462no—
313C2×C8—1462no—
314C2×C8—1462no—
315C2×C8—1462no—
316C2×C8—1462no—
317C2×C8—1462no—
318C2×C8—1462no—
319C4⋊C4—24no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—2921no—
324C4⋊C4—2921no—
325C4⋊C4—1464no—
326C4⋊C4—1464no—
327C22⋊C4—1464no—
328C22⋊C4—1464no—
329C42—22no—
330C42—734no—
331C42—1462no—
332C42—1462no—
333C42—1462no—
334C42—1462no—
335C42—1462no—
336C42—1462no—
337C23—732no—
338C23—1461no—
339C23—1461no—
340C23—1461no—
341C23—1461no—
342C23—1461no—
343C23—1461no—
344C23C23×#6440811yescharacteristic
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—2924no—
358C2×C4—2924no—
359C2×C4—732no—
360C2×C4—731no—
361C2×C4—731no—
362C2×C4—732no—
363C2×C4—732no—
364C2×C4—1464no—
365C2×C4—1464no—
366C2×C4—1461no—
367C2×C4—1462no—
368C2×C4—1462no—
369C2×C4—1462no—
370C2×C4—1462no—
371C2×C4—1461no—
372C2×C4—1462no—
373C2×C4—1462no—
374C2×C4—1461no—
375C2×C4—1461no—
376C2×C4—1461no—
377C2×C4#1455667012yes—
378C2×C4D4×#6440812yes—
379C2×C4#1455667912yes—
380C8—2922no—
381C8—2922no—
382C8—2922no—
383C8—1464no—
384C22—2921no—
385C22—2921no—
386C22—2921no—
387C22—2921no—
388C22—1462no—
389C22—1462no—
390C22—1462no—
391C22C2×#1455667011yescharacteristic
392C22C2×D4×#6440811yescharacteristic
393C22#1798965511yescharacteristic
394C22C2×#1455667911yescharacteristic
395C22#1798966511yescharacteristic
396C22#1798966511yescharacteristic
397C22#1798966711yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—42no—
402C4—42no—
403C4—2921no—
404C4—2922no—
405C4—2922no—
406C4—732no—
407C4—732no—
408C4—1462no—
409C2—2922no—
410C2#2262643911yescharacteristic
411C2#2262661311yescharacteristic
412C2#2262662211yescharacteristic
413C2#2262666011yescharacteristic
414C2#2262667711yescharacteristic
415C2#2262683911yescharacteristic
416C2#2262684511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#384×C73C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenter#384#64408
hypocenterC73#22602
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #22626033C2semidirect
3. #22625988C2semidirect
4. #22625987C2semidirect
7. #22625731C2semidirect
218. C73#22602semidirect

Maximal subgroups

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

subgroupindex
C2×#179896122
#226260332
#226259882
#226259872
#226258252
#226257882
#226257312
#226256582
#226123362
#2260273