Finite Group Databasebeta

← #10695540

Subgroups — #10695540

Order 768.

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

normal subgroups only

This group has 568 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
1#78642C211yesmaximal, characteristic
2#78625C211yesmaximal, characteristic, complemented
3#74033C211yesmaximal, characteristic
4#69977C211yesmaximal, characteristic
5#69975C211yesmaximal, characteristic, complemented
6#69529C211yesmaximal, characteristic, complemented
7C3×#1305C211yesmaximal, characteristic, complemented
8#9936—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5134—22no—
17#5121—21no—
18#5120—21no—
19#4988—21no—
20#4984—21no—
21#4961—22no—
22#4738—21no—
23#4736—21no—
24#4731—21no—
25#4723—22no—
26#4722—21no—
27#4722—21no—
28#4721—21no—
29#4677—21no—
30#4674—21no—
31#4555—22no—
32#4527—22no—
33#4470—22no—
34#3193—31no—
35#3095—31no—
36#2624—31no—
37#1996—31no—
38#1826—31no—
39#1507—31no—
40#1305D311yescharacteristic, complemented
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C2×C3×M16D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C22⋊C4D412yes—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59#926—21no—
60#926—21no—
61#926—21no—
62#926—21no—
63C2×C4⋊Dic3—22no—
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D411yescharacteristic
67C2×C4⋊Dic3D411yescharacteristic
68#924—21no—
69#924—21no—
70#924—21no—
71#924—21no—
72C2×Dic3⋊C4—21no—
73C2×C4×Dic3—21no—
74C2×C4×Dic3—21no—
75C2×#204—22no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8D412yes—
80C3×#95—21no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83#821—22no—
84#818—22no—
85#818—22no—
86#817—22no—
87#816—22no—
88#816—22no—
89#816—22no—
90C2×C22⋊Q8—31no—
91C2×#121—31no—
92C2×#117—31no—
93#485—62no—
94#484—61no—
95#484—61no—
96#483—61no—
97#480—61no—
98#468—62no—
99#450—61no—
100#448—61no—
101#428—61no—
102#419—61no—
103#407C2×D311yescharacteristic
104#407C2×D311yescharacteristic
105#399—61no—
106#397—61no—
107#394—62no—
108#390—61no—
109#389—61no—
110#380C2×D311yescharacteristic
111C4×C22⋊C4—31no—
112#342—62no—
113#339—62no—
114#327—62no—
115C24×C3C2×D411yescharacteristic
116C23×D3—21no—
117C22×C3×C4—22no—
118C22×C3×C4—42no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—82no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—41no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×M16—42no—
141C3×M16—42no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C3×C4⋊C4—42no—
145C3×C22⋊C4—42no—
146#217—22no—
147#217—41no—
148#217—41no—
149#217—41no—
150#217—41no—
151#217—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—82no—
155(C2×D3)⋊C4—82no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—82no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—41no—
168C4×Dic3—41no—
169C4×Dic3—42no—
170#204—82no—
171C22×Q8—31no—
172C22×D4—31no—
173C22×D4—61no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C2×M16—21no—
177C2×M16D1212yes—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#117—61no—
187#117—61no—
188#117—61no—
189#117—61no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4D1212yes—
204C2×C42—61no—
205C2×C42—61no—
206#107—62no—
207#107—62no—
208#106—62no—
209#103—62no—
210#103—62no—
211#103—62no—
212#102—62no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8D1212yes—
217#95—21no—
218#95—62no—
219#95—61no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3#12011yescharacteristic
225C23×C3#12111yescharacteristic
226C23×C3#12711yescharacteristic
227C22×D3—42no—
228C22×D3—42no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245#67—82no—
246#67—82no—
247C2×Dic3—41no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C2×Dic3—81no—
262D12—42no—
263D12—81no—
264Dic6—42no—
265Dic6—42no—
266C3×C8—42no—
267C3×C8—42no—
268C24—61no—
269C24C2×D1211yescharacteristic
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—122no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—122no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—242no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—121no—
287C22×C4C2×D1211yescharacteristic
288C22×C4C2×D1211yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23911yescharacteristic
293SD16—242no—
294M16—42no—
295M16—42no—
296C2×C8—22no—
297C2×C8—22no—
298C4⋊C4—42no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4—242no—
310C22⋊C4—42no—
311C22⋊C4—62no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—242no—
320C22⋊C4—242no—
321C42—122no—
322C42—121no—
323C42—121no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#39211yescharacteristic
335C22×C3#45511yescharacteristic
336C22×C3#45611yescharacteristic
337C2×D3—82no—
338C2×D3—82no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4#45812yes—
347Dic3—82no—
348Dic3—81no—
349Dic3—81no—
350Dic3—81no—
351Dic3—81no—
352Dic3—81no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—122no—
358C23—122no—
359C23—121no—
360C23—121no—
361C23—121no—
362C23#87411yescharacteristic
363C23#88211yescharacteristic
364C23#89511yescharacteristic
365Q8—122no—
366Q8—122no—
367D4—122no—
368D4—241no—
369D4—242no—
370D4—242no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4#88211yescharacteristic
396C2×C4#88211yescharacteristic
397C2×C4#89511yescharacteristic
398C2×C4#89511yescharacteristic
399C8—42no—
400C8—42no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3#200711yescharacteristic
406C2×C3#200811yescharacteristic
407C2×C3#202211yescharacteristic
408D3—161no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—242no—
420C22—242no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22#467111yescharacteristic
425C22#474611yescharacteristic
426C22#475211yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—242no—
432C4—241no—
433C4—241no—
434C4—241no—
435C4—241no—
436C4—241no—
437C4#474212yes—
438C3#993611yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—481no—
444C2#6896611yescharacteristic
445C2#6900911yescharacteristic
446C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1305D3
hypocenterC3#9936
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
2. #78625C2semidirect
5. #69975C2semidirect
6. #69529C2semidirect
7. C3×#1305C2semidirect
40. #1305D3semidirect
438. C3#9936semidirect

Maximal subgroups

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

subgroupindex
#786422
#786252
#740332
#699772
#699752
#695292
C3×#13052
#99363