Finite Group Databasebeta

← #32982148

Subgroups — #32982148

Order 18688.

594 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988864C211yesmaximal, characteristic
2C2×#17988652C211yesmaximal, characteristic
3#22610734C211yesmaximal, characteristic, complemented
4#22609932C211yesmaximal, characteristic
5#22607815C211yesmaximal, characteristic, complemented
6#22607765C211yesmaximal, characteristic, complemented
7C2×#332×C73C211yesmaximal, characteristic, complemented
8C22×C8×D73—21no—
9C2×C4×D292C2211yescharacteristic
10#17988864—22no—
11#17988864C414yescomplemented
12#17988652—22no—
13#17988652—22no—
14C2×#14556524—21no—
15C2×C4⋊#10596831C2211yescharacteristic
16C2×#14556515—21no—
17C2×#14556515—21no—
18C2×#14556513—21no—
19C2×#14556511—21no—
20C2×#14556503—21no—
21C2×C4×#10596831C2211yescharacteristic
22#17988457C2211yescharacteristic
23#17988350—21no—
24C2×C4⋊Q8×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27#332×C73—22no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31C22×#10596831D411yescharacteristic
32C22×#10596831D411yescharacteristic
33C2×C8×D73—24no—
34C2×C8×D73—42no—
35C2×C8×D73—42no—
36C4×D292—21no—
37C4×D292—21no—
38C4×D292C2×C412yes—
39C4×D292D412yes—
40C2×C4⋊Dic73—21no—
41C2×C4⋊Dic73—21no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73D411yescharacteristic
44C2×#11924005—21no—
45#14556524—24no—
46C4⋊#10596831—21no—
47C4⋊#10596831C2×C412yes—
48#14556515—42no—
49#14556515—42no—
50#14556513—42no—
51#14556511—24no—
52#14556503—24no—
53C4×#10596831—21no—
54C4×#10596831—21no—
55C22×Q8×C73—21no—
56C22×C8×C73—21no—
57C22×C8×C73—21no—
58C4⋊Q8×C73—21no—
59C4⋊Q8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73C2×C412yes—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C42×C73C2311yescharacteristic
65C4⋊C8×C73—21no—
66C4⋊C8×C73—21no—
67C4⋊C8×C73—21no—
68C4⋊C8×C73D412yes—
69C23×D73—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292C22⋊C412yes—
74C2×D292SD1612yes—
75C2×D292Q1612yes—
76C22×Dic73—21no—
77C22×Dic73—41no—
78C2×C4×D73—22no—
79C2×C4×D73—22no—
80C2×C4×D73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×#10596831—21no—
85C2×#10596831—21no—
86C2×#10596831—22no—
87C2×#10596831—21no—
88C2×#10596831—21no—
89C2×#10596831—21no—
90C2×#10596831—41no—
91C2×#10596831—41no—
92C2×#10596831D812yes—
93C8×D73—48no—
94C4⋊Dic73—21no—
95C4⋊Dic73—41no—
96C4⋊Dic73—41no—
97C4⋊Dic73—41no—
98C4⋊Dic73—41no—
99C4⋊Dic73—81no—
100C4⋊Dic73C22⋊C412yes—
101#11924005—22no—
102#11924005—22no—
103C2×Q8×C73—22no—
104C2×Q8×C73—41no—
105C2×Q8×C73—41no—
106C2×Q8×C73—41no—
107C2×Q8×C73—41no—
108C2×Q8×C73—41no—
109C22×C4×C73—41no—
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73#4111yescharacteristic
113C2×C8×C73—22no—
114C2×C8×C73—41no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73—41no—
118C2×C8×C73—41no—
119C4⋊C4×C73—22no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—41no—
123C42×C73C22×C411yescharacteristic
124C42×C73C2×D411yescharacteristic
125C42×C73C2×D411yescharacteristic
126C42×C73#4111yescharacteristic
127C23×C73#12111yescharacteristic
128C22×D73—22no—
129C22×D73—22no—
130C22×D73—22no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135Q8×C73—42no—
136Q8×C73—42no—
137Q8×C73—81no—
138Q8×C73—81no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—42no—
142C2×C4×C73—81no—
143C2×C4×C73—81no—
144C2×C4×C73C2×C22⋊C411yescharacteristic
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73#12011yescharacteristic
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73#12311yescharacteristic
150C2×C4×C73C2×D811yescharacteristic
151C2×C4×C73C2×SD1611yescharacteristic
152C2×C4×C73C2×Q1611yescharacteristic
153C2×C4×C73#13511yescharacteristic
154C2×C4×C73#13511yescharacteristic
155C2×C4×C73#13511yescharacteristic
156C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
158C2×Dic73—22no—
159C2×Dic73—41no—
160C2×Dic73—41no—
161C2×Dic73—81no—
162C2×Dic73—81no—
163C2×Dic73—81no—
164D292—22no—
165D292—22no—
166D292—22no—
167D292#10314yescomplemented
168C4×D73—42no—
169C4×D73—42no—
170C4×D73—42no—
171C4×D73—42no—
172C8×C73—42no—
173C8×C73—81no—
174#10596831—22no—
175#10596831—42no—
176#10596831—41no—
177C22×C73#38611yescharacteristic
178C22×C73#45911yescharacteristic
179C22×C73#46311yescharacteristic
180C22×C73#46511yescharacteristic
181C22×C73#46711yescharacteristic
182C22×C73#47111yescharacteristic
183C22×C73#47311yescharacteristic
184C2×D73—24no—
185C2×D73—42no—
186C2×D73—42no—
187C2×D73—42no—
188C2×D73—42no—
189C2×D73—42no—
190C2×D73—42no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—82no—
194C4×C73C2×#10311yescharacteristic
195C4×C73#41611yescharacteristic
196C4×C73C4×D811yescharacteristic
197C4×C73#44211yescharacteristic
198C4×C73#44711yescharacteristic
199C4×C73#44911yescharacteristic
200C4×C73#46511yescharacteristic
201C4×C73#46711yescharacteristic
202Dic73—42no—
203Dic73—161no—
204#9219—731nomaximal, Sylow
205C2×C73#166511yescharacteristic
206C2×C73#166911yescharacteristic
207C2×C73#167911yescharacteristic
208C2×C73#168311yescharacteristic
209C2×C73#187911yescharacteristic
210C2×C73#189211yescharacteristic
211C2×C73#190811yescharacteristic
212D73—44no—
213C2×#474—731no—
214C2×C8×D4—731no—
215#1956—731no—
216#1802—731no—
217C2×#332D7311yescharacteristic, complemented
218#1296—731no—
219#1277—731no—
220C73#921911yescharacteristic, complemented, Sylow
221C23×C8—1461no—
222C2×C4⋊Q8C2×D7311yescharacteristic
223C2×C4×D4—731no—
224#474—734no—
225#474—1462no—
226C8×D4—1462no—
227C8×D4—1462no—
228C2×#107—1461no—
229C2×C4⋊C8—731no—
230C2×C4⋊C8C2×D7311yescharacteristic
231C2×C4⋊C8C2×D7311yescharacteristic
232C2×#103—1461no—
233C2×#103—1461no—
234C2×#102—1461no—
235C2×#102—1461no—
236C2×C22⋊C8—1461no—
237C2×C4×C8—731no—
238#384—731no—
239#340—1461no—
240#332—22no—
241C22×Q8—21no—
242C22×D4—731no—
243C23×C4—1461no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—2922no—
247C22×C8—2922no—
248C22×C8—731no—
249C22×C8—731no—
250C22×C8—1464no—
251C22×C8—1461no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8C4×D7312yes—
256C4×D4—732no—
257C4×D4—732no—
258C4×D4—1461no—
259C4×D4—1461no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—731no—
263C2×C4⋊C4—1461no—
264C2×C4⋊C4—1461no—
265C2×C4⋊C4—1461no—
266C2×C22⋊C4—1461no—
267C2×C42C22×D7311yescharacteristic
268#107—1464no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—732no—
273C4⋊C8—1461no—
274C4⋊C8C22⋊D7312yes—
275#103—2922no—
276#103—2922no—
277#102—2922no—
278#102—1464no—
279C22⋊C8—1464no—
280C4×C8—1461no—
281C4×C8—1461no—
282C24—1461no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—732no—
290C2×D4—732no—
291C2×D4—732no—
292C2×D4—1461no—
293C2×D4—1461no—
294C2×D4—1461no—
295C22×C4—41no—
296C22×C4—2921no—
297C22×C4—2921no—
298C22×C4—2921no—
299C22×C4—2921no—
300C22×C4—2921no—
301C22×C4—1461no—
302C22×C4—1462no—
303C22×C4—1462no—
304C22×C4D4×D7311yescharacteristic
305C22×C4D4×D7311yescharacteristic
306C22×C4#1192402811yescharacteristic
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—2928no—
314C2×C8—2921no—
315C2×C8—2921no—
316C2×C8—732no—
317C2×C8—1462no—
318C2×C8—1461no—
319C2×C8—1461no—
320C2×C8—1461no—
321C2×C8—1461no—
322C2×C8—1461no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—2921no—
328C4⋊C4—2921no—
329C4⋊C4—2921no—
330C4⋊C4—2921no—
331C4⋊C4—5841no—
332C4⋊C4—732no—
333C4⋊C4—1461no—
334C22⋊C4—1462no—
335C22⋊C4—1462no—
336C42C2×C4×D7311yescharacteristic
337C42D4×D7311yescharacteristic
338C42C2×C22⋊D7311yescharacteristic
339C42#1192402711yescharacteristic
340C23—2921no—
341C23—2921no—
342C23—2921no—
343C23—2921no—
344C23—1462no—
345C23—1462no—
346C23—1462no—
347C23#1455656411yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352D4—734no—
353D4—1462no—
354D4—1462no—
355D4—1462no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—2921no—
362C2×C4—2921no—
363C2×C4—2922no—
364C2×C4—2922no—
365C2×C4—2922no—
366C2×C4—2922no—
367C2×C4—5841no—
368C2×C4—5841no—
369C2×C4—5841no—
370C2×C4—1462no—
371C2×C4C22⋊C4×D7311yescharacteristic
372C2×C4#1455655111yescharacteristic
373C2×C4C4×C22⋊D7311yescharacteristic
374C2×C4#1455655911yescharacteristic
375C2×C4#1455656211yescharacteristic
376C2×C4#1455657311yescharacteristic
377C2×C4D8×D7311yescharacteristic
378C2×C4SD16×D7311yescharacteristic
379C2×C4Q16×D7311yescharacteristic
380C2×C4#1455661211yescharacteristic
381C2×C4#1455661211yescharacteristic
382C2×C4#1455661411yescharacteristic
383C2×C4#1455661511yescharacteristic
384C2×C4#1455661811yescharacteristic
385C8—42no—
386C8—81no—
387C8—2921no—
388C8—2922no—
389C8—1462no—
390C22—2922no—
391C22—2922no—
392C22—2922no—
393C22—2922no—
394C22—2922no—
395C22—2922no—
396C22—1464no—
397C22#1798846311yescharacteristic
398C22#1798878111yescharacteristic
399C22#1798880111yescharacteristic
400C22#1798881911yescharacteristic
401C22#1798882711yescharacteristic
402C22#1798885311yescharacteristic
403C22#1798886111yescharacteristic
404C4—21no—
405C4—21no—
406C4—82no—
407C4—2922no—
408C4—11681no—
409C4#103×D7311yescharacteristic
410C4#1798858211yescharacteristic
411C4#1798866711yescharacteristic
412C4#1798869611yescharacteristic
413C4#1798871911yescharacteristic
414C4#1798873111yescharacteristic
415C4#1798881311yescharacteristic
416C4#1798882511yescharacteristic
417C2—2924no—
418C2#2260927511yescharacteristic
419C2#2260929511yescharacteristic
420C2#2260935311yescharacteristic
421C2#2260938111yescharacteristic
422C2#2261027611yescharacteristic
423C2#2261035111yescharacteristic
424C2#2261044311yescharacteristic
subgroupG/H
centerC23#14556564
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC2×#332×C73C2
socleC23×C73#121
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenterC2×#332D73
hypocenterC73#9219
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #22610734C2semidirect
5. #22607815C2semidirect
6. #22607765C2semidirect
7. C2×#332×C73C2semidirect
11. #17988864C4semidirect
167. D292#103semidirect
217. C2×#332D73semidirect
220. C73#9219semidirect

Maximal subgroups

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

subgroupindex
C2×#179888642
C2×#179886522
#226107342
#226099322
#226078152
#226077652
C2×#332×C732
#921973