Finite Group Databasebeta

← #10696411

Subgroups — #10696411

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2541C211yesmaximal, characteristic
2C2×#5070C211yesmaximal, characteristic, complemented
3C2×#5007C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#69486C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#10070—31nomaximal, Sylow
9C2×C3×#127—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×#121—22no—
12C2×C3×C4×D4C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C3×#392—22no—
15C3×#390—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17C2×#832—21no—
18C2×#832—22no—
19#5070—22no—
20#5070—22no—
21#5007—22no—
22#5007—22no—
23C2×#810—22no—
24C2×#807—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4544—22no—
29D4⋊(C3⋊C8)—22no—
30C2×#486—31no—
31C2×#474—31no—
32#2541D311yescharacteristic
33#1981—31no—
34C2×#331—31no—
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×D4—24no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C22×C3×D4D412yes—
42C23×C3×C4—21no—
43C22×Dic6—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—42no—
49C3×#127—42no—
50C3×#121—44no—
51C3×#121—44no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D412yes—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869D412yescomplemented
71C3×#95—21no—
72#832—24no—
73#832—28no—
74#810—44no—
75#807—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yescomplemented
80C4×C3⋊C8—21no—
81C4×C3⋊C8D412yescomplemented
82C2×C4⋊Q8—31no—
83C2×#127—21no—
84C2×#127C2×D311yescharacteristic
85C2×#121—22no—
86C2×C4×D4C2×D311yescharacteristic
87#486—62no—
88#486—62no—
89#474—62no—
90#474—62no—
91C2×#107—61no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8—31no—
94C2×#103—62no—
95C2×#102—62no—
96C2×#102—61no—
97C2×C4×C8—31no—
98#392—22no—
99#390—21no—
100C4×C4⋊C4C2×D311yescharacteristic
101#331—62no—
102D4⋊C8—62no—
103C24×C3—21no—
104C24×C3—22no—
105C2×C3×D4—21no—
106C2×C3×D4—22no—
107C2×C3×D4—21no—
108C2×C3×D4—22no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—44no—
113C2×C3×D4—48no—
114C2×C3×D4—48no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—42no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—84no—
126C2×C3×D4SD1612yes—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×Dic3—41no—
137C2×Dic6—22no—
138C2×Dic6—42no—
139C2×Dic6—42no—
140C2×Dic6—41no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4D814yes—
145C3×C22⋊C4—48no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—42no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—42no—
165C22×Q8—61no—
166C22×D4—24no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4#6711yescharacteristic
170C22×D4#6712yes—
171C23×C4—21no—
172C22×C8—61no—
173C22×C8—61no—
174C22×C8—61no—
175C4⋊Q8—32no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127—21no—
183#127—42no—
184#127—42no—
185#121—44no—
186#121—44no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4#6711yescharacteristic
194C2×C4⋊C4#6712yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42C22×D311yescharacteristic
199#107—64no—
200C4⋊C8—32no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—124no—
205#102—64no—
206#102—68no—
207C4×C8—32no—
208C4×C8—61no—
209#95—21no—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—44no—
220C23×C3#12011yescharacteristic
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—44no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—82no—
227C3×D4—816no—
228C3×D4—81no—
229C3×D4—81no—
230C3×D4—82no—
231C3×D4—81no—
232C3×D4—81no—
233C3×D4—81no—
234C3×D4—81no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—82no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12711yescharacteristic
250C2×C3×C4C2×D812yes—
251C2×C3×C4C2×SD1611yescharacteristic
252C2×C3×C4#13511yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4AGL(1,ℤ/8)12yes—
255C2×C3×C4#13711yescharacteristic
256C2×Dic3—42no—
257C2×Dic3—82no—
258Dic6—44no—
259Dic6—81no—
260Dic6—81no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263C3⋊C8—81no—
264C24—21no—
265C24—22no—
266C2×Q8—62no—
267C2×Q8—121no—
268C2×Q8—122no—
269C2×Q8—122no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—44no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—84no—
291C2×D4#21412yes—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—121no—
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—122no—
305C2×C8—121no—
306C2×C8—121no—
307C2×C8—122no—
308C2×C8—121no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C4⋊C4#21314yes—
317C22⋊C4—48no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42—41no—
321C42—41no—
322C42D3×D411yescharacteristic
323C42D3×D411yescharacteristic
324C42D3×D411yescharacteristic
325C42#23711yescharacteristic
326C22×C3—44no—
327C22×C3—42no—
328C22×C3—82no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—84no—
332C22×C3—84no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3—82no—
336C22×C3#39211yescharacteristic
337C22×C3#44712yes—
338C22×C3#45011yescharacteristic
339C22×C3#45211yescharacteristic
340C22×C3#45512yes—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—44no—
344C3×C4—81no—
345C3×C4#44811yescharacteristic
346C3×C4#44911yescharacteristic
347C3×C4#45912yes—
348C3×C4#46511yescharacteristic
349C3×C4#46811yescharacteristic
350C3×C4#49311yescharacteristic
351C3×C4#49611yescharacteristic
352Dic3—82no—
353C23—22no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—44no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—44no—
363C23#95311yescharacteristic
364Q8—124no—
365Q8—241no—
366Q8—241no—
367D4—22no—
368D4—22no—
369D4—44no—
370D4—41no—
371D4—41no—
372D4—82no—
373D4—816no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—82no—
379D4—81no—
380D4—81no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—122no—
391C2×C4—242no—
392C2×C4C2×#21312yes—
393C2×C4#93211yescharacteristic
394C2×C4#93212yes—
395C2×C4C2×#21411yescharacteristic
396C2×C4#93711yescharacteristic
397C2×C4#93711yescharacteristic
398C2×C4#93811yescharacteristic
399C2×C4#93911yescharacteristic
400C2×C4#93911yescharacteristic
401C2×C4#94011yescharacteristic
402C2×C4#95011yescharacteristic
403C2×C4#95111yescharacteristic
404C8—122no—
405C8—122no—
406C8—241no—
407C2×C3—82no—
408C2×C3—162no—
409C2×C3#163912yes—
410C2×C3#164112yes—
411C2×C3#200011yescharacteristic
412C2×C3#200411yescharacteristic
413C2×C3#201111yescharacteristic
414C22—42no—
415C22—44no—
416C22—81no—
417C22—81no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—84no—
423C22—84no—
424C22#521712yes—
425C22#521811yescharacteristic
426C22#522311yescharacteristic
427C22#523911yescharacteristic
428C22#524212yes—
429C4—21no—
430C4—21no—
431C4—44no—
432C4—81no—
433C4—242no—
434C4#503711yescharacteristic
435C4#504011yescharacteristic
436C4#504111yescharacteristic
437C4#504211yescharacteristic
438C4#507212yes—
439C4#507311yescharacteristic
440C4#507411yescharacteristic
441C3#1007011yescharacteristic, complemented, Sylow
442C2—82no—
443C2—162no—
444C2#7168511yescharacteristic
445C2#7176711yescharacteristic
446C2#7185711yescharacteristic
447C2#7190812yes—
448C2#7192012yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2541C2
socleC23×C3#120
hypercenter#2541D3
hypocenterC3#10070
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. C2×#5070C2semidirect
3. C2×#5007C2semidirect
4. C2×#4544C2semidirect
5. C2×D4⋊(C3⋊C8)C2semidirect
6. #69486C2semidirect
7. #67619C2semidirect
70. #869D4semidirect
79. C4⋊(C3⋊C8)D4semidirect
81. C4×C3⋊C8D4semidirect
441. C3#10070semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
C2×#50702
C2×#50072
C2×#45442
C2×D4⋊(C3⋊C8)2
#694862
#676192
#100703