Finite Group Databasebeta

← #10695977

Subgroups — #10695977

Order 768.

566 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3109C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic, complemented
3#82383C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#70812C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#10005—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#481—21no—
13C3×#476—21no—
14C3×#459—21no—
15C3×#447—21no—
16C3×#442—22no—
17C3×#432C2211yescharacteristic
18C2×#837—22no—
19#5222—21no—
20#5212—21no—
21#5212—21no—
22#5081—21no—
23#5071—21no—
24#5034—21no—
25#5033—21no—
26#5009—22no—
27#4985—21no—
28#4945—22no—
29C4×C4⋊Dic3C2211yescharacteristic
30#4926C2211yescharacteristic
31C4×#208C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4465—22no—
34#3234—31no—
35#3109D311yescharacteristic
36#2562—31no—
37#1980—31no—
38#1528—31no—
39#1490—31no—
40#1285—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×D8—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60C2×#217—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3D412yes—
67C2×#208—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71#870—21no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×#102—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×#97D412yes—
78#837—44no—
79#832—22no—
80#832—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84#830—21no—
85#808—22no—
86#807—22no—
87C4⋊(C3⋊C8)D412yes—
88#803—21no—
89#803—21no—
90C4×C3⋊C8—21no—
91#532C2×D311yescharacteristic
92C2×#125—31no—
93C2×C4×D4C2×D311yescharacteristic
94#501—61no—
95#490—61no—
96#481—21no—
97#481—61no—
98#480—61no—
99#478—62no—
100#476—21no—
101#459—21no—
102#447—21no—
103#442—22no—
104#432—31no—
105#432C2×D311yescharacteristic
106C2×#104—62no—
107C4×M16—31no—
108#400—61no—
109#394—61no—
110#388—62no—
111#380—61no—
112#380—61no—
113C4×C4⋊C4—31no—
114#330—62no—
115D4⋊C8—62no—
116C24×C3—21no—
117C3×#41—42no—
118C3×#41—42no—
119C2×C3×Q8—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×Dic3—22no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C3×D8—44no—
140C2×C3×C8—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#217—44no—
154#217—82no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—42no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C4×Dic3—42no—
165#208—22no—
166#208—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×#41—21no—
170C22×D4#6711yescharacteristic
171C23×C4—21no—
172C2×D8—22no—
173C2×M16—61no—
174C4⋊Q8#6711yescharacteristic
175#127#6711yescharacteristic
176#125—61no—
177#125—61no—
178#125—61no—
179#125—61no—
180#124—21no—
181#121—21no—
182#121—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C42—32no—
199C2×C42C22×D311yescharacteristic
200#107—62no—
201#106—22no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205#104—124no—
206#102—22no—
207#102—22no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212#97—61no—
213#97—61no—
214#97#6712yes—
215C4×C8—61no—
216#95—62no—
217#95—62no—
218#95—61no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3#12011yescharacteristic
228C3×Q8—42no—
229C3×D4—22no—
230C3×D4—22no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—81no—
237C3×D4—81no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—82no—
247C2×C3×C4—82no—
248C2×C3×C4—82no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12711yescharacteristic
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C3×C8—42no—
264C3⋊C8—42no—
265C3⋊C8—42no—
266C24—21no—
267#41—42no—
268#41—42no—
269C2×Q8—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—42no—
276C2×D4—41no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—62no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4C2×#6711yescharacteristic
289D8—44no—
290M16—62no—
291M16—121no—
292C2×C8—22no—
293C2×C8—62no—
294C2×C8—121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—124no—
313C22⋊C4—242no—
314C42—122no—
315C42D3×D411yescharacteristic
316C42#23711yescharacteristic
317C42C2×#6711yescharacteristic
318C42C2×#6711yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3#39211yescharacteristic
332C22×C3#45311yescharacteristic
333C22×C3#45411yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—81no—
340C3×C4—81no—
341C3×C4#45412yes—
342Dic3—82no—
343Dic3—81no—
344Dic3—81no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23#93711yescharacteristic
354Q8—42no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—81no—
363D4—81no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4#93711yescharacteristic
384C2×C4#93811yescharacteristic
385C2×C4#93911yescharacteristic
386C2×C4#93911yescharacteristic
387C2×C4#94011yescharacteristic
388C2×C4#95311yescharacteristic
389C8—42no—
390C8—122no—
391C8—122no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—82no—
395C2×C3—161no—
396C2×C3#200311yescharacteristic
397C2×C3#200611yescharacteristic
398C2×C3#201011yescharacteristic
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22#519811yescharacteristic
412C22#520111yescharacteristic
413C22#522311yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—81no—
421C4—242no—
422C4—241no—
423C4—241no—
424C4#520112yes—
425C3#1000511yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2#7171511yescharacteristic
431C2#7209611yescharacteristic
432C2#7209811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3109C2
socleC22×C3#392
hypercenter#3109D3
hypocenterC3#10005
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
2. #83273C2semidirect
3. #82383C2semidirect
4. #71818C2semidirect
5. #71599C2semidirect
6. #70812C2semidirect
7. #67749C2semidirect
425. C3#10005semidirect

Maximal subgroups

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

subgroupindex
C3×#31092
#832732
#823832
#718182
#715992
#708122
#677492
#100053