Finite Group Databasebeta

← #10697603

Subgroups — #10697603

Order 768.

740 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2541C211yesmaximal, characteristic
2#83169C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69471C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10246—31nomaximal, Sylow
9C2×C3×#127—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×#121—22no—
12C2×C3×C4×D4C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#392—22no—
15C3×#390—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17#5218—22no—
18#5215—21no—
19#5076—21no—
20#5075—22no—
21#5070—21no—
22#5009—22no—
23#5008—21no—
24#5007—21no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4544—22no—
30D4⋊(C3⋊C8)—22no—
31#4452—22no—
32#3208—31no—
33#3077—31no—
34#2541D311yescharacteristic
35#1545—31no—
36#1503—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×D4—24no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C22×C3×D4D412yes—
44C23×C3×C4—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—42no—
50C3×#127—42no—
51C3×#121—48no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#214—24no—
63#926—21no—
64#924—21no—
65#924—21no—
66C2×#208—21no—
67#870—21no—
68#869D411yescharacteristic, complemented
69#869D411yescharacteristic, complemented
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#95D412yes—
73#832—22no—
74#832—22no—
75#832—24no—
76#830—21no—
77#830—21no—
78#808—22no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yescomplemented
83C4⋊(C3⋊C8)D412yescomplemented
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86#533—31no—
87C2×#127—21no—
88C2×#127C2×D311yescharacteristic
89C2×#121—22no—
90C2×C4×D4C2×D311yescharacteristic
91#492—61no—
92#486—61no—
93#478—62no—
94#476—61no—
95#474—61no—
96#460—62no—
97#450—62no—
98#431—31no—
99#431—31no—
100#428—61no—
101#423—31no—
102#418—61no—
103#392—22no—
104#390—21no—
105C4×C4⋊C4C2×D311yescharacteristic
106#331—62no—
107#328—62no—
108D4⋊C8—62no—
109C24×C3—21no—
110C24×C3—22no—
111C2×C3×D4—21no—
112C2×C3×D4—22no—
113C2×C3×D4—21no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—44no—
119C2×C3×D4—48no—
120C2×C3×D4—48no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—82no—
130C2×C3×D4—84no—
131C2×C3×D4SD1612yescomplemented
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—44no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C2×Dic6—21no—
141C2×Dic6—21no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
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
154#217—41no—
155#214—84no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C22×D4—24no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4#6711yescharacteristic
172C22×D4#6712yes—
173C23×C4—21no—
174C2×SD16—64no—
175C2×M16—61no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127—42no—
183#127—42no—
184#125—61no—
185C22⋊Q8—61no—
186C22⋊Q8—61no—
187#121—48no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194#117—61no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4#6711yescharacteristic
197C2×C22⋊C4—24no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—62no—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#102—62no—
208#102—62no—
209#102—64no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C4×C8—61no—
213C4×C8—61no—
214#95#6712yes—
215C23×C3—22no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—44no—
219C23×C3—44no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3#12011yescharacteristic
226C3×D4—22no—
227C3×D4—22no—
228C3×D4—41no—
229C3×D4—44no—
230C3×D4—41no—
231C3×D4—82no—
232C3×D4—82no—
233C3×D4—82no—
234C3×D4—816no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C3×D4—81no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—82no—
247C2×C3×C4—82no—
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12711yescharacteristic
254C2×C3×C4C2×SD1611yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
257C2×C3×C4#13711yescharacteristic
258C2×Dic3—41no—
259C2×Dic3—41no—
260Dic6—42no—
261Dic6—81no—
262C3⋊C8—42no—
263C3⋊C8—42no—
264C3⋊C8—81no—
265C24—21no—
266C24—22no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C2×D4—84no—
289C2×D4#21412yescomplemented
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—44no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4C2×#6711yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—244no—
299M16—121no—
300M16—121no—
301C2×C8—62no—
302C2×C8—62no—
303C2×C8—121no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—48no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—121no—
317C42—41no—
318C42—41no—
319C42—121no—
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C42#23711yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—44no—
327C22×C3—42no—
328C22×C3—82no—
329C22×C3—84no—
330C22×C3—84no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3—82no—
336C22×C3#39211yescharacteristic
337C22×C3#45011yescharacteristic
338C22×C3#45211yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—44no—
343C3×C4—81no—
344C3×C4#44811yescharacteristic
345C3×C4#44911yescharacteristic
346C3×C4#45312yes—
347C3×C4#46511yescharacteristic
348C3×C4#46811yescharacteristic
349Dic3—81no—
350Dic3—81no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—44no—
355C23—44no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23#95311yescharacteristic
362Q8—122no—
363Q8—241no—
364D4—22no—
365D4—22no—
366D4—41no—
367D4—41no—
368D4—44no—
369D4—82no—
370D4—82no—
371D4—82no—
372D4—816no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4#93211yescharacteristic
389C2×C4C2×#21411yescharacteristic
390C2×C4#93711yescharacteristic
391C2×C4#93711yescharacteristic
392C2×C4#93811yescharacteristic
393C2×C4#93911yescharacteristic
394C2×C4#93911yescharacteristic
395C2×C4#94011yescharacteristic
396C2×C4#95011yescharacteristic
397C2×C4#95111yescharacteristic
398C8—122no—
399C8—122no—
400C8—241no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—82no—
404C2×C3—162no—
405C2×C3#200011yescharacteristic
406C2×C3#200211yescharacteristic
407C2×C3#200811yescharacteristic
408C22—21no—
409C22—21no—
410C22—44no—
411C22—42no—
412C22—82no—
413C22—84no—
414C22—84no—
415C22—81no—
416C22—81no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22#521811yescharacteristic
421C22#522311yescharacteristic
422C22#523911yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—44no—
427C4—81no—
428C4—241no—
429C4—241no—
430C4#503711yescharacteristic
431C4#504011yescharacteristic
432C4#504111yescharacteristic
433C4#504211yescharacteristic
434C4#507712yes—
435C3#1024611yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—82no—
439C2—162no—
440C2#7168511yescharacteristic
441C2#7176911yescharacteristic
442C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2541C2
socleC22×C3#392
hypercenter#2541D3
hypocenterC3#10246
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. #83169C2semidirect
3. #82296C2semidirect
4. #71759C2semidirect
5. #71597C2semidirect
6. #69471C2semidirect
7. #67620C2semidirect
68. #869D4semidirect
69. #869D4semidirect
82. C4⋊(C3⋊C8)D4semidirect
83. C4⋊(C3⋊C8)D4semidirect
131. C2×C3×D4SD16semidirect
289. C2×D4#214semidirect
435. C3#10246semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
#831692
#822962
#717592
#715972
#694712
#676202
#102463